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d4281958079281c89405631f9ca4db59ed09d74d | subsection | 54 | 74 | Error analysis | With regularities (REF ), it is straightforward to check that, for any 1 \le n \le N , the weak solution (c,\mu ,{{v}},p) to model problem (REF ) satisfies\big ({\partial _t c}(t^n),\chi \big ) + a_{\mathcal {D}}(\mu ^n,\chi ) + a_{\mathcal {A}}(c^n,{{v}}^n,\chi ) = 0, \quad \forall \chi \in S_h,\\
\big (\Phi _{+}\,\!^... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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b6f83d4ab09d40949d438b20f92c4c70eb44028e | subsection | 55 | 74 | Error analysis | In addition, there exists a constant C_1>0 independent of mesh size h, such that|(\lambda , \phi )| \le C_1 \Vert \phi \Vert _{\mathrm {DG}} \Vert \mathcal {J}(\lambda )\Vert _{\mathrm {DG}}, &&
\forall \phi \in H^1(\mathcal {T}_h)\,,\quad \forall \lambda \in M_h.The linearity of the operator \mathcal {J} is easy to ch... | {
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Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
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1dde68fdad71c544e195ffb2b2cdef8010ffc610 | subsection | 56 | 74 | Error analysis | There is an approximation operator \mathcal {R}_h: H_0^1(\mathcal {T}_h)^d \rightarrow \mathbf {X}_h
satisfyingb_\mathcal {P}(\phi ,\mathcal {R}_h({{v}})-{{v}}) = 0,\quad \forall {{v}}\in H_0^1(\mathcal {T}_h)^d, \quad \forall \phi \in Q_h,and for all E in \mathcal {T}_h, for all {{v}} in H_0^1(\mathcal {T}_h)^d \cap W... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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f72bb88905c8011d8fb4b1c4738a55a031a4f03e | subsection | 57 | 74 | Error analysis | We also have for all s, 1 \le s \le q+1,\forall {{v}}\in H_0^1(\mathcal {T}_h)^d \cap H^s(\Omega )^d,\quad \Vert \mathcal {R}_h({{v}})-{{v}}\Vert _{\mathrm {DG}} \le C h^{s-1} \vert {{v}}\vert _{H^s(\Omega )}.With the operator \mathcal {R}_h, we have a bound for the form a_\mathcal {C} (see Proposition 6.2 in ).
[Bound... | {
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Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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de7a2861e0d0940706f587f4e29c0c9514c9299e | subsection | 58 | 74 | Error analysis | There exist a constant C, independent of mesh size h and time step size \tau , such that for all 0 \le n \le N\Vert c^n - \mathcal {P}_hc^n\Vert _{\mathrm {DG}} &\le Ch^q \Vert c\Vert _{L^\infty (0,T;\,H^{q+1}(\Omega ))},\\
\Vert \mu ^n - \mathcal {P}_h\mu ^n\Vert _{\mathrm {DG}} &\le Ch^q \Vert \mu \Vert _{L^\infty (0... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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9aaba2d1fb2620f77698bceff42a8b79959854bb | subsection | 59 | 74 | Error analysis | We note that for all n \ge 1a_{\mathcal {D}}(\zeta _\mu ^n,\chi )=0, \quad b_{\mathcal {P}}(\phi ,{\zeta }_{{{v}}}^n)=0, && \forall \chi \in S_h, \quad \forall \phi \in Q_h.Therefore, from (REF ) and (REF ), the error equation becomes, for any \chi \in S_h , \varphi \in S_h , {{\theta }} \in \mathbf {X}_h , and \phi \i... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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5e7b57d4ac485712accf639780468a4ca06a70f9 | subsection | 60 | 74 | Error analysis | Suppose (c,\mu ,{{v}},p) is a weak solution of (REF ) with regularity (REF ). Then, under as:CHNS:assumptionA and sufficiently small time step size \tau , there exists a constant C independent of mesh size h and time step size \tau such that for any m \ge 1\max _{1\le n\le m}\Big (\Vert \xi _c^n \Vert _{\mathrm {DG}}^2... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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c9a10ceea40b362866b8297bfaee11856f84a53b | subsection | 61 | 74 | Error analysis | \end{aligned}Choosing \varphi = \delta _\tau \xi _c^n in () and adding and subtracting the appropriate terms, we obtain\begin{aligned}\kappa a_{\mathcal {D}}(\xi _c^n,\delta _\tau \xi _c^n) - (\xi _\mu ^n,\delta _\tau \xi _c^n)
= (\zeta _\mu ^n,\delta _\tau \xi _c^n) + \big (\Phi _+\,\!^{\prime }(c_h^n)-\Phi _+\,\!^{\p... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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f209df7808cdcb3712018ce93124d9b35e5cf178 | subsection | 62 | 74 | Error analysis | \end{aligned}Summing (REF ) – (REF ), we obtain the following equation\begin{aligned}a_{\mathcal {D}}\big (\mathcal {J}(\delta _\tau \xi _c^n),\mathcal {J}(\delta _\tau \xi _c^n)\big ) + \kappa a_{\mathcal {D}}(\xi _c^n,\delta _\tau \xi _c^n) + \mu _\mathrm {s} a_{{\varepsilon }}({\xi }_{{{v}}}^n, {\xi }_{{{v}}}^n) + (... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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61fe588accd229050f006345e7d18648541751ab | subsection | 63 | 74 | Error analysis | \end{aligned}The remainder of the proof consists of finding lower bounds for the terms in the left-hand side and upper bounds for the terms in the right-hand side of the equation above. We will then utilize Gronwall's lemma.
For the left-hand side of (REF ), since a_{\mathcal {D}} and the inner product are both symmetr... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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efed813cef8f138bd134049d35ae2047219af3b2 | subsection | 64 | 74 | Error analysis | By Cauchy–Schwarz's inequality, Poincaré's inequality, Young's inequality, and using a Taylor expansion, we haveT_1 &\le \Vert \delta _\tau c^n - (\partial _t c)^n\Vert _{L^2(\Omega )}\Vert \mathcal {J}(\delta _\tau \xi _c^n)\Vert _{L^2(\Omega )} \\
&\le C_P \Vert \delta _\tau c^n - (\partial _t c)^n\Vert _{L^2(\Omega ... | {
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"raw": "B. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation, Frontiers in Applied Mathematics, Society for Industrial and Applied Mathematics,... | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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ecc545677266154b505ffe97c0cf34dc87b3ec57 | subsection | 65 | 74 | Error analysis | We give an outline of the proof for completeness.\begin{split}
T_{11} + T_{12}
=& - a_{\mathcal {C}}({{v}}_h^{n-1},{{v}}^n,{{v}}^n,{\xi }_{{{v}}}^n)
+ a_{\mathcal {C}}({{v}}_h^{n-1},{{v}}_h^{n-1},{{v}}_h^n,{\xi }_{{{v}}}^n)\\
=& -a_{\mathcal {C}}({{v}}_h^{n-1},{{v}}_h^{n-1},{\xi }_{{{v}}}^n,{\xi }_{{{v}}}^n)
- a_{\math... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.007792978547513485,
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6dbb5e80f23499290c08189d443966f2f837f946 | subsection | 66 | 74 | Error analysis | We rewrite the second term as:T_{\mathcal {C}}^2 = -a_{\mathcal {C}}({{v}}_h^{n-1},{\xi }_{{{v}}}^{n-1},{{v}}^n, {\xi }_{{{v}}}^n)
+ a_{\mathcal {C}}({{v}}_h^{n-1},{\xi }_{{{v}}}^{n-1},{\zeta }_{{{v}}}^n, {\xi }_{{{v}}}^n).Note that {\xi }_{{{v}}}^{n-1} belongs to V_h and we apply lem:CHNS:boundconvection to the first ... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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-... | |
db02ec2c81da4126c8cd19fd300d86712c3bb978 | subsection | 67 | 74 | Error analysis | The term simplifiesT_\mathcal {C}^4 = \sum _{E\in \mathcal {T}_h} \int _E \left(({{v}}^{n-1}-{{v}}^n)\cdot \nabla \mathcal {R}_h{{v}}^n\right) \cdot {\xi }_{{{v}}}^n
\\
+ \sum _{E\in \mathcal {T}_h} \int _{\partial E_-\setminus \partial \Omega } \vert ({{v}}^{n-1}-{{v}}^n)\cdot {{n}}_E\vert \left((\mathcal {R}_h{{v}}^n... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.014054429717361927,
-0.005... | |
466a5bc975f42a31297ccae89ba3022b384f7322 | subsection | 68 | 74 | Error analysis | Using Poincaré's inequality, we obtain:|a_\mathcal {A}(\zeta _c^{n-1},{\xi }_{{{v}}}^n,\mu ^n)| \le \Vert \zeta _c^{n-1}\Vert _{L^2(\Omega )} \Vert {\xi }_{{{v}}}^n\Vert _{L^4(\Omega )} \Vert \mu ^n \Vert _{W^{1,4}(\Omega )}
\le C h^q \Vert c^{n-1}\Vert _{H^{q+1}(\Omega )} \Vert {\xi }_{{{v}}}^n\Vert _{\mathrm {DG}},|a... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.048311639577150345,
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-... | |
804cc382ca2444b0f0534db9b3cb89610677ace1 | subsection | 69 | 74 | Error analysis | We note that \delta _\tau c^n - (\partial _t c)^n - \delta _\tau \zeta _c^n belongs to M_h by taking \chi = 1 in (REF ). We choose \chi = \xi _\mu ^n in (REF ), use coercivity of a_\mathcal {D}, lem:CHNS:errorpropertyofJ, Cauchy–Schwarz's inequality, triangular inequality, and Poincaré's inequality to obtain:K_\alpha \... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.03793569654226303,
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0.017... | |
57d8c35636603915fbc38987db6c9311b2a9bff3 | subsection | 70 | 74 | Error analysis | Using Young's inequality, we have\Vert \xi _\mu ^n\Vert _{\mathrm {DG}}^2
\le C\Big ( \Vert {\xi }_{{v}}^n\Vert _{L^2(\Omega )}^2 + \mu _\mathrm {s}K_{{\varepsilon }}\Vert {\xi }_{{v}}^n\Vert _{\mathrm {DG}}^2
+ \Vert \xi _c^{n-1}\Vert _{\mathrm {DG}}^2 + \tau ^2 + h^{2q}\\
+ \Vert J(\delta _\tau \xi _c^n)\Vert _{\math... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.06587234139442444,
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... | |
cfe743bd52cc1a6c15b81ea881b3f2304cc82036 | subsection | 71 | 74 | Error analysis | \Vert \partial _{tt} c\Vert _{L^2(\Omega )}^2.To this end, combining (REF ) with all the bounds for T_1 to T_{16}, and choosing the values r_1 = K_\alpha /9, r_2 = 1/18, and r_3 = \min {\lbrace 1,K_\alpha \rbrace }/18C yields&\frac{K_\alpha }{2}\Vert \mathcal {J}(\delta _\tau \xi _c^n)\Vert _{\mathrm {DG}}^2
+ \frac{\k... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.... | |
b35f426651e010796460a2cd873a65b706099b18 | subsection | 72 | 74 | Error analysis | Then, under as:CHNS:assumptionA and sufficiently small time step size \tau , there exists a constant C independent of mesh size h and time step size \tau such that for any m \ge 1\max _{1\le n\le m}\Big (\Vert c(t^n)-c_h^n \Vert _{\mathrm {DG}}^2
+ \Vert {{v}}(t^n)-{{v}}_h^n\Vert _{L^2(\Omega )}^2\Big )
+ \tau \sum _{n... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.010459101758897305,
0.002441219985485077,
-0.054469719529151917,
0.00290276319719851,
-0.0119314631447196,
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0.00013898742327000946,
0.0052867671474814415,
-0.03326162323355675,
0.002500343369320035,
... | |
667b4fd5c20c27bf1a82802acb5012e2608ef7dc | subsection | 73 | 74 | Conclusions | In this paper, we have formulated an interior penalty discontinuous Galerkin method for
solving the Cahn–Hilliard–Navier–Stokes equations. The time discretization utilizes a convex-concave
splitting of the chemical energy density and a Picard's linearization for the convection term. Existence and uniqueness of the nume... | {
"cite_spans": []
} | 1807.02725 | Numerical analysis of a discontinuous Galerkin method for
Cahn-Hilliard-Navier-Stokes equations | [
"Chen Liu",
"Beatrice Riviere"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.007711320649832487,
-0.011835980229079723,
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0.04078109562397003,
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-0.0016025505028665066,
-0.0222678221762180... | |
8b3696458ef8778513ffd55254d00281165496ee | abstract | 0 | 84 | Abstract | We introduce an approximation technique for nonlinear hyperbolic systems with
sources that is invariant domain preserving. The method is
discretization-independent provided elementary symmetry and skew-symmetry
properties are satisfied by the scheme. The method is formally first-order
accurate in space. A series of hig... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.021026192232966423,
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0.02662605606019497,
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-0.012130496092140675,
0.0030993798281997442,
0... |
4114c4c7c0098a4a2c7fb482915737c46863abd6 | subsection | 1 | 84 | Introduction | The present paper is
concerned with the approximation of hyperbolic systems in conservation
form with a source term:{\left\lbrace \begin{array}{ll} \partial _t + ()=(),
\quad \mbox{for}\, (,t)\in _+,\\
(,0) = _0(), \quad \mbox{for}\, \in ^d.
\end{array}\right.}The space dimension d is arbitrary. The dependent variable
... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.03167073056101799,
0.009946683421730995,
-0.02425648458302021,
0.02248682826757431,
-0.011289181187748909,
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0.029153548181056976,
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0.013302926905453205,
0.05626283586025238,
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0.006662905216217041,
0.0006421675207093358,
0.... |
90a0e27c3b055271e682e7b2d618c427157c5795 | subsection | 2 | 84 | Introduction | A brief overview of
explicit Runge Kutta Strong Stability methods is made in
§. The key result of this section is a
reformulation of the Shu-Osher Theorem REF which does
not involve any norm. We show therein that only convexity matters. It
seems that the result, as reformulated, is not well known in the
literature. We ... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.04854240268468857,
0.00290804379619658,
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0.017513098195195198,
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0.025781476870179176,
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... |
e3bd04e20455588f02297f54a8f2077c11ee43db | subsection | 3 | 84 | Preliminaries | We recall in this section key properties about the
system (REF ) that will be used repeatedly in the
paper. The reader who is familiar with hyperbolic systems with source
terms, Riemann problems, and invariant sets is invited to jump to
§. | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.021718360483646393,
0.020283697172999382,
-0.02959374524652958,
0.03568343445658684,
-0.014934234321117401,
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0.06526191532611847,
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0.008394306525588036,
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0.013652194291353226,
-0.006299545522779226,
0.... |
20b08131382a63359edee0cabc5582973fdd0152 | subsection | 4 | 84 | Riemann problem space average and maximum wave speed | We consider (REF ) without source term in this
subsection, () = 0. Instead of trying to give a precise meaning to
the solutions of (REF ), which is either a very technical
task or a completely open problem, we instead assume that there
is a clear notion of solution for the Riemann problem.
That is to say
we assume that... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.03821749985218048,
0.02855628728866577,
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0.00031502917408943176,
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0.030174121260643005,
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0.03205141797661781,
0.006620144005864859,
0.0... |
82c637eb966182fe90d643d6608b30d034fdf168 | subsection | 5 | 84 | Invariant sets and invariant domains | We introduce in this section the notions of invariant sets and
invariant domains. Our definitions are slightly different from those
in
ChuehConleySmoller1977,Hoff1985,Smoller1994,Frid2001. We
associate invariant sets with solutions of Riemann problems
and define invariant domains only for an approximation process; our ... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.05022554472088814,
-0.0006121810292825103,
-0.04897448420524597,
0.02340400591492653,
0.0022904344368726015,
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0.04015602543950081,
0.043359965085983276,
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0.017667431384325027,
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0.041986845433712006,
-0.024700837209820747,
0.... |
bc301790fb5626d3038ff8ef3c733e0fa777af4b | subsection | 6 | 84 | Examples | We briefly go over some examples of systems with
source terms and show that the proposed definition for invariant sets is
meaningful/useful. | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.003936609718948603,
0.01930159516632557,
-0.04885668680071831,
0.05669938772916794,
-0.01505982130765915,
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0.04537782073020935,
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0.018355587497353554,
0.006026980467140675,
0.005515831522643566,
-0.0009464838658459485,
-0.... |
f2edeb91e0820cd54294808bce7ebff70fa50a2e | subsection | 7 | 84 | Euler + co-volume EOS | For the compressible Euler
equations with covolume of state the dependent variable is
=(\rho ,,E), where \rho is the density, is the
momentum, and E is the total energy. The flux is
()=(\rho , \otimes + p , (E + p)) where
:= /\rho and the pressure is given by the equation of state
p ( 1 - b \rho ) = ( \gamma - 1 ) e \r... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.008857128210365772,
-0.007045269478112459,
-0.011550122871994972,
0.007323723752051592,
-0.008613004349172115,
-0.04400337487459183,
0.03301778808236122,
0.017775287851691246,
0.01002434641122818,
0.02220003679394722,
-0.011771360412240028,
0.0023783030919730663,
-0.008727437816560268,
... |
dc0fa9684409e890fdaf8a4188b982b9a5d3b2c1 | subsection | 8 | 84 | Shallow water | Saint-Venant's shallow water model describes the time and space
evolution of a body of water evolving in time under the action of
gravity assuming that the deformations of the free surface are small
compared to the water elevation and the bottom topography z varies
slowly. The dependent variable is =(,), where
is the ... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.074335016310215,
0.014059876091778278,
-0.028791086748242378,
0.00003236279371776618,
0.001068031881004572,
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0.0019663230050355196,
0.028119752183556557,
0.019758589565753937,
0.00768982945010066,
-0.015326256863772869,
-0.012984215281903744,
-0.004214911255985498,
... |
9e23fd2a271e44de166333f42857e5947682358a | subsection | 9 | 84 | ZND model | We now consider the Zel'dovich–von Neumann–Döring model for compressible
reacting flows. The dependent variable is
=(\rho _1,\rho _2,, E), where \rho _1 is the density of
the burned gas (fuel), \rho _2 is the density of the unburned gas, is
the momentum of the mixture, and E is the total energy. The flux is
()=(\rho _1... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.011744186282157898,
-0.005646977107971907,
-0.009073318913578987,
0.03369871899485588,
-0.008798601105809212,
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0.03201988711953163,
0.007215158082544804,
0.030661558732390404,
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-0.004929658491164446,
-0.015071324072778225,
-0... |
fa54c1c76c700acb6c6237d1108f540fc70850b6 | subsection | 10 | 84 | ZND model | Moreover,
\rho _2 -\tau \kappa (T)\rho _2 =\rho _2(1 -\tau \kappa (T)) \ge \rho _2(1
-\tau \kappa _0);
hence \rho _2 -\tau \kappa (T)\rho _2\ge 0 provided
\tau \le \tau _0:=\kappa _0^{-1}. Finally, observing that
\rho :=\rho _1 +\tau \kappa (T)\rho _2 + \rho _2 -\tau \kappa (T)\rho _2 >0,
we have
\rho e(+ \tau ()) = E ... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
0.0333264134824276,
0.02517792396247387,
0.011391103267669678,
0.059297826141119,
0.026032445952296257,
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0.022324426099658012,
0.012733925133943558,
-0.024231843650341034,
-0.033356934785842896,
0.017975... |
6f9f9beebf157d86e79dd28bc5245909d0a0394f | subsection | 11 | 84 | Euler equations with sources | In some astrophysical
applications one may want to solve the compressible Euler equations
with Coriolis effects, gravitation effects and some heat transfer
effects due to the emission and/or absorption of radiation. The
dependent variables and the flux are the same as those of Euler's
equations, but the source term is
... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.04857566952705383,
0.00672416016459465,
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0.010107217356562614,
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-0.0006626902031712234,
-0.02109929360449314,
-0... |
e2b701ebad7ed504e4c5758006d5977f43ed4e27 | subsection | 12 | 84 | Abstract low-order approximation | In this section
we describe a generic invariant domain preserving technique for
approximating solutions to (REF ). In order to
stay general we present the method without referring to any particular
discretization technique, we are going to use instead the graph
theoretic language to describe the method. The method is i... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.05253618210554123,
-0.00040781707502901554,
0.02152397111058235,
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0.023812131956219673,
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0.039325863122940063,
-0.0172832440584898,
0.04... |
ad20e86b00fb11495d4e2e09d3a2d9630ec5d936 | subsection | 13 | 84 | The low-order scheme | To
identify properly the time stepping technique, we denote by t^n the
current time, n\in , and we denote by the current time
step size; that is t^{n+1}:=t^n+. We now address the approximation in
space by assuming that we have at hand some finite-dimensional vector
space X_h with some basis \lbrace \varphi \rbrace _{i\... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.00962909311056137,
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0.006912803743034601,
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0.04068329930305481,
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0.03305327519774437,
-0.002605654066428542,
0.0... |
31d976ed0126835f2c85b5c4cdb5ac381c2a3729 | subsection | 14 | 84 | The low-order scheme | We
assume that the graph viscosity \lbrace d_{ij}\rbrace _{(i,j)\in } is scalar and has
the following properties:d_{ij}= d_{ji}> 0, \quad \text{if} \quad i\ne j.Although the diagonal value d_{ii} is not needed, we adopt the convention
d_{ii}:=-\sum _{j\in (i)\backslash \lbrace i\rbrace } d_{ij}. This convention will
he... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.043514594435691833,
-0.004886237438768148,
-0.03216296061873436,
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0.012312861159443855,
0.025937872007489204,
0.045101381838321686,
-0.04806135222315788,
-0.001804207917302847,
-0.00661034323275089... |
492aa35e2d34aebd96e879b13f25ea82560fe9cb | subsection | 15 | 84 | The low-order scheme | Note that if all the values
\lbrace _j\rbrace _{j\in (i)} are constant, the graph viscosity term
\sum _{j\in (i)} d_{ij}(^n_j - ^n_i)
vanishes; which in some sense implies that
(REF ) is a first-order consistent
perturbation of (REF ). The scalars m_i
and the vectors \lbrace _{ij}\rbrace _{j\in (i)} are not uniquely
d... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.034181077033281326,
0.0007324516773223877,
-0.0475178025662899,
-0.023011188954114914,
0.011696337722241879,
0.015854526311159134,
-0.0030213631689548492,
0.010574771091341972,
0.011749745346605778,
0.03387589007616043,
-0.04660223796963692,
0.0016823498299345374,
-0.01716683618724346,
... |
92089f938e620f5f8fe33a54a3c0ad9e815d2e5c | subsection | 16 | 84 | Invariant domain preserving graph viscosity | Now we propose a definition of the graph viscosity that makes the
algorithm (REF ) invariant domain preserving. Recall that the discretization
setting is still unspecified. Most of
the arguments presented in this subsection are generalizations of those
in §3.2, §4.1 and §4.2 of GuePo2016.Since \sum _{j\in (i)} (_i^n)_{... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.07281192392110825,
-0.007518456317484379,
0.000054893600463401526,
-0.022627845406532288,
0.01690603606402874,
0.009696558117866516,
-0.004100629594177008,
0.029494015499949455,
0.017302749678492546,
0.03622286394238472,
-0.04723925143480301,
0.013213562779128551,
-0.00958212185651064,
... |
4697e5d53f877fc402f9a23759108dfd67cc5347 | subsection | 17 | 84 | Invariant domain preserving graph viscosity | The state
\overline{}_{ij}^{n} defined in (REF ), with
d_{ij} as defined in (REF ), belongs to .Let us set
t_{ij}:= \Vert _{ij}\Vert _{\ell ^2}/(2d_{ij}), then according to
Lemma REF , we have
\overline{}_{ij}^{n}:=\overline{}(t_{ij},_{ij},_{i}^n,_{j}^n)
\in
if
\lambda _{\max }(_{ij},_{i}^n,_{j}^n) t_{ij}\le \frac{1}{... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.03510455787181854,
0.025168441236019135,
-0.05308419466018677,
-0.018895410001277924,
0.0005485087167471647,
0.06666813045740128,
-0.013042106293141842,
0.05925038829445839,
0.004033208359032869,
0.011340298689901829,
-0.029609931632876396,
0.037821345031261444,
-0.01819331757724285,
0.... |
1df10d699473ecc49672db015adc733bc6887f74 | subsection | 18 | 84 | Invariant domain preserving graph viscosity | But we have established in Lemma REF that
\overline{}_{ij}^{n}\in . Then, the
convexity of implies _1 is in . Since is an
invariant set according to Definition REF and
_i^{n}\in by assumption, the condition 2\le \tau _0
implies that _2:= _{i}^{n} + 2(_i^n) is a
member of . In conclusion, the convexity of implies that
_... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.026389308273792267,
0.022863611578941345,
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0.016789034008979797,
0.008638720959424973,
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0.0015052895760163665,
-0.015163550153374672,
... |
aebe4320bf7cfcb5eea19147a2fed3f761f0b3fe | subsection | 19 | 84 | Invariant domain preserving graph viscosity | Then recalling
(REF ), the CFL condition and the convexity
of \eta imply that\eta (_i- (_i^n)) \le \Big (1 - \sum _{j\in (i)\backslash \lbrace i\rbrace } \frac{2d_{ij}}{m_i} \Big )\eta (_i^n)
+ \sum _{j\in (i)\backslash \lbrace i\rbrace }\frac{2d_{ij}}{m_i}\eta (\overline{}_{ij}^{n}).Lemma REF implies that
\eta (\overl... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.05344251170754433,
0.028140489012002945,
-0.034794095903635025,
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0.0033897538669407368,
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0.02930029295384884,
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0.019060451537370682,
-0.01950300671160221,
0.061133839190006256,
-0.021639486774802208,... |
700fe6433459d11ae5b31f8700bcc0d1891f9809 | subsection | 20 | 84 | Invariant domain preserving graph viscosity | In this case one has to come up
with some informed guess. We now give a lower bound on
\lambda _{\max }(,_L,_R) that guaranties positivity if
it happens that some components of , say , has to be
positive (think of the density and the total energy in the Euler
equations or the water height in the shallow water equations... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.057575713843107224,
0.04936804994940758,
-0.026636777445673943,
0.010068885050714016,
-0.007490640506148338,
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0.03173224255442619,
0.028360692784190178,
-0.017986690625548363,
0.02469928003847599,
-0.003060712246224284,
... |
cf8a0154579c037f6f125e7f94c66c539c147154 | subsection | 21 | 84 | Examples of
discretizations | In this section
we illustrate the GMS-GV scheme described in § in
the following three space discretization settings: finite volumes,
continuous finite elements, and discontinuous elements. | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
0.003536369651556015,
-0.007732708938419819,
0.007232963107526302,
-0.040895234793424606,
-0.019806722179055214,
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0.02832147479057312,
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0.0015011450741440058,
0.00046278946683742106,
0.010803665965795517,
... |
022c3d1a425ee444764a1de85eeaeaa3069dcc2f | subsection | 22 | 84 | Finite Volumes | We now illustrate the construction of the abstract low-order
scheme (REF )–(REF ) in the context of finite volumes
(FV). | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.011213981546461582,
-0.0034595513716340065,
0.0023801103234291077,
0.015104546211659908,
0.006415618117898703,
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0.0054506054148077965,
0.046747807413339615,
-0.012419293634593487,
0.0022523319348692894,
-0.0062897466123104095,... |
3b2b40ea5f066c9314ebad360dc78458e33322db | subsection | 23 | 84 | Technical preliminaries | We unify our presentation by putting into a single framework the
so-called cell-centered and vertex-centered finite volume techniques,
see Figure REF . We refer the reader to
Barth2004,Eymard2000 for comprehensive reviews on the finite
volume techniques. For any manifold E\subset ^d of dimension
l we denote by |E| the ... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
0.016294747591018677,
0.015325911343097687,
-0.046107422560453415,
0.004439861048012972,
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0.03246743604540825,
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0.05031842365860939,
0.01193117257207632,
0.023526685312390327,
0.011099652387201786,
0.05... |
0f58b9b1101f6cc07d2b7e5d8eab88f3bb776908 | subsection | 24 | 84 | Technical preliminaries | Similarly, letting
\partial K to be the boundary of the cell K, we denote by
(\partial K) the set of indices of the shape functions with
non-vanishing trace on \partial K:(K) := \big \lbrace i \in \varphi _{i|K} \lnot \equiv 0 \big \rbrace ,\qquad (\partial K) := \big \lbrace i \in \varphi _{i|\partial K} \lnot \equiv ... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.008743672631680965,
0.027497554197907448,
-0.06824032217264175,
-0.030305296182632446,
-0.019577892497181892,
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0.04214663803577423,
0.02075287140905857,
-0.0012493686517700553,
-0.003177783451974392,
0.... |
0b1f3194f3cfd7bc44c6ac1878a21248afe58ea8 | subsection | 25 | 84 | Definitions of | We define the connectivity graph (,) by identifying the
vertices of this graph with the cells in _h, and we say that a
pair of cells K_i,K_j form an edge of the graph, (i,j)\in , iff the cells K_i and K_j share an interface, \partial K_i\cap \partial K_j is a (d-1)-manifold of positive
measure. For any i\in we define t... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.008697551675140858,
0.0244752150028944,
-0.03503434732556343,
-0.03048720583319664,
-0.0246735792607069,
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0.07348667830228806,
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0.009880113415420055,
0.00040316773811355233,
0.... |
a5c4858a14563b9540d0fc5cf1995bb8e67968a9 | subsection | 26 | 84 | Definitions of | Let us mention in passing that while any
family of vectors of the form
_{ij} =\alpha _{ij} |\Gamma _{ij}| satisfies the conservation
constraint (REF ), only the factor \alpha = \frac{1}{2}
leads to a consistent discretization of the divergence operator.We define the connectivity graph (,) by identifying the
shape funct... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.013225079514086246,
-0.0018334249034523964,
-0.03299783170223236,
0.003125026123598218,
-0.03087632544338703,
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0.048107851296663284,
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0.008524185977876186,
-0.00220735976472497,
... |
c51647be43d55831a93f64f056f225f7c3eb33e9 | subsection | 27 | 84 | Definitions of | On the other hand, the skew-symmetry property _{ij} = -_{ji}
follows using integration by parts if is the d-torus (which
is the case for periodic boundary conditions) or if either \varphi _i
or \varphi _j vanish at the boundary of D (which is the case when
we solve the Cauchy problem).We start by defining the undirecte... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.019181106239557266,
0.014877946116030216,
-0.019944077357649803,
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0.07306215912103653,
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0.019928818568587303,
... |
22e3f01e0dea04658c3a56d0882cfa4e2013c803 | subsection | 28 | 84 | Definitions of | Inserting (REF )
into (REF ) and integrating by parts, we obtain\int _K \nabla ((_h)) \varphi _i\approx \int _K ((_h)) \varphi _i\\
+\int _{\partial K} \tfrac{1}{2}((_h^\mathsf {e})-(_h^\mathsf {i}))_K \varphi _i s
+ \int _{\partial K} \alpha _{\partial K}^n (_h^\mathsf {i} -_h^\mathsf {e}) \varphi _i s.We now consider... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.025043733417987823,
0.027073480188846588,
-0.03201812878251076,
-0.020266957581043243,
-0.022434057667851448,
0.04004555568099022,
0.03217074275016785,
0.02792811207473278,
-0.003086590440943837,
0.05179672688245773,
-0.014185343869030476,
-0.00039631593972444534,
0.010225046426057816,
... |
8f70b55ccaef6200c0f597595e23050506ffe2e9 | subsection | 29 | 84 | Definitions of | \end{array}\right.}Therefore, (REF ) can be rewritten as follows:\begin{aligned}\int _K \nabla ((_h)) \varphi _i\approx \sum _{j\in (i)} (_j^n)_{ij} + \int _{\partial K} \alpha _{\partial K}^n (_h^i -_h^e) \varphi _i s.
\end{aligned}The set of coefficients \lbrace _{ij}\rbrace _{j \in (i)}
defined in (REF ) satisfy the... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.007724211551249027,
0.04644444212317467,
-0.04003620892763138,
-0.0005745477974414825,
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0.018919549882411957,
0.014525331556797028,
0.020475834608078003,
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0.022856036201119423,
-0.03286508843302727,
0.017454810440540314,
-0.031186740845441818,
... |
5bbf66db264706299deb02843d83d66c46696cc1 | subsection | 30 | 84 | Definitions of | The partition of
unity property on \partial K (see (REF ))
implies that
\sum _{j \in (\partial K)} _{ij}^{\partial } =
\int _{\partial K} \varphi _i _K s
and
\sum _{j \in (\partial K)} _{ij}^{\partial } =
\int _{\partial K} \varphi _i _K s;
hence, the last two summations cancel each other. This completes the proof. | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.0007147453143261373,
0.059018224477767944,
-0.019942300394177437,
0.00014697815640829504,
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0.008002858608961105,
-0.020003333687782288,
0.009559182450175285,
-0.0023554649669677... |
ffe0f69f316ad2afec3f6beb81d39c4d56f4401a | subsection | 31 | 84 | Continuous finite elements | We describe in this section one possible implementation of the
abstract low-order scheme (REF )–(REF ) in
the context of continuous finite elements (cG). The set of the
d-variate polynomials of degree at most k\in is denoted
_{k,d}. The reader who is familiar with
GuePo2016,guermondpopovsecondorder2018,GuerNazPopTom201... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.0557868592441082,
-0.012191930785775185,
-0.02059963345527649,
-0.03360028937458992,
-0.0040016695857048035,
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0.02044704370200634,
0.007137391250580549,
-0.003929189406335354,
0.031174110248684883,
0.0015888420166447759,
0.0184175968170166,
-0.004901949781924486,
0.... |
c81d559675e4df12394e9b71748a5f9909d342d9 | subsection | 32 | 84 | Discontinuous finite elements | We finally describe in this section one possible implementation of the
abstract low-order scheme (REF )–(REF ) in the context of
discontinuous finite elements (dG). This section builds on
top of the definitions and notation already introduced in
§REF . | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.016400648280978203,
-0.003211475908756256,
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-0.023311804980039597,
-0.014074045233428478,
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0.05501464009284973,
0.026530910283327103,
-0.0015552010154351592,
0.04030745476484299,
-0.01853654719889164,
-0.002420049160718918,
0.001790722017176449... |
dbeb5972808e19c0fea5f9fc24721d7e7e536f9b | subsection | 33 | 84 | Graph viscosity for dG | It is important to notice at this stage, that the formulation of the
viscous fluxes
\int _{\partial K} \alpha _{\partial K}^n (_h^{n,\textup {i}}
-_h^{n,\textup {e}}) \varphi _i s in (REF ) is not
compatible with our pursuit of a purely algebraic formulation. Note that
the dissipation in (REF ) is active only on \parti... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
0.0105596287176013,
0.01423718687146902,
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0.048952728509902954,
0.01306982897222042,
0.005142477806657553,
0.04852546006441116,
-0.00025726694730110466,
-0.017151765525341034,
0.0001257723051821813,
0.0... |
5e8abe1a921ec38ed5cf3358269d0376051908e3 | subsection | 34 | 84 | Runge Kutta SSP time integration | Increasing the time accuracy while keeping the invariant domain
property can be done by using so-called Strong Stability Preserving
(SSP) time discretization methods. The key idea is to achieve
higher-order accuracy in time by making convex combinations of forward
Euler steps. More precisely each time step of a SSP met... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.03182481974363327,
0.00892498530447483,
-0.03658480942249298,
0.009664920158684254,
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0.007979089394211769,
0.03277071565389633,
0.050040941685438156,
-0.008062999695539474,
0.0353948138654232,
-0.028865080326795578,
-0.002... |
e59d2035c355bd385f37b4c5a76f607306c20a0f | subsection | 35 | 84 | SSPRK methods | We are going to illustrate the SSP concept with explicit Runge Kutta
methods. Let us consider a finite-dimensional vector space E, a
subset A\subset E and a (nonlinear) operator
L:[0,T]A\longrightarrow E. We are interested in
approximating in time the following problem
\partial _t u +L(t,u)=0 with appropriate initial c... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.06327501684427261,
0.04667828977108002,
-0.04637320339679718,
0.021417098119854927,
-0.02799171954393387,
0.03246124088764191,
0.01571197248995304,
0.009213623590767384,
0.03142394497990608,
0.039691802114248276,
0.02253066375851631,
0.051468152552843094,
-0.006319111678749323,
0.007169... |
f6cdcf56c47fb4e863161f08200e93d8f11a0996 | subsection | 36 | 84 | SSPRK methods | Any Runge Kutta
method that admits an (\alpha -\beta ) representation as defined above
is said to be SSP for a reason that will be stated in
Theorem REF .[Midpoint rule]
The midpoint rule, defined by the Butcher tableau\begin{array}{c|ccccc}
0 \\
\frac{1}{2} & \frac{1}{2} \\[2pt] \hline & 0 & 1
\end{array}does not have... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.06208077445626259,
-0.0009437826811335981,
-0.028781795874238014,
0.0017883704276755452,
-0.03174238279461861,
0.017534613609313965,
0.003839988959953189,
0.04398151487112045,
0.011827089823782444,
0.01191102433949709,
-0.00505894236266613,
0.04907860606908798,
-0.010453620925545692,
0.... |
b4aacc04a83f2252592c8eaf5dfa2b154b718391 | subsection | 37 | 84 | SSPRK methods | \end{aligned}[SSPRK(3,3), SSPRK(4,3)] The following Runge–Kutta methods, which are
third-order and composed of three substeps and four substeps, respectively, are SSP:\begin{array}{|ccc|ccc|c|c|}
\hline &\alpha & & & \beta & &\gamma & c_{\textup {os}} \\ \hline \hline 1 & & & 1 & & &0 & \\
\frac{3}{4}&\frac{1}{4}& & 0 ... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.03772854804992676,
0.05118994414806366,
-0.04746593162417412,
-0.007207647897303104,
-0.013781907968223095,
0.036934904754161835,
0.01741435006260872,
-0.0160712618380785,
-0.002024170011281967,
0.0035752071999013424,
-0.004704621620476246,
0.053418248891830444,
-0.0040101842023432255,
... |
f782a37941463de9c464ce1e8d0c2a8c95b60fff | subsection | 38 | 84 | The key result | We
henceforth denotec_\textup {os}:=\inf _{\lbrace \alpha _{ik}\ne 0,\ 1\le k+1\le i\le s\rbrace }\alpha _{ik}\beta _{ik}^{-1}.The following theorem is the main result of this section.
[Shu-Osher] Assume that the Runge Kutta method
with the Butcher tableau (REF ) is SSP. Let B\subset A be convex.
Let u^n\in B and assum... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.04331319034099579,
0.026677750051021576,
-0.04981474578380585,
-0.005116542335599661,
-0.008249041624367237,
0.039161961525678635,
0.01181268785148859,
0.008592433296144009,
0.03336245194077492,
0.016269154846668243,
-0.006513003259897232,
0.032080452889204025,
-0.024876849725842476,
0.... |
f3d6ab6df8096669a0eb75ada1863070c30c91f2 | subsection | 39 | 84 | The key result | The
assumption (REF ) then consists of stating
that + L(t,\cdot ) maps any ball B centered at 0 into B for any
s \in [0,_{\max }] and any t\in [0,T]. In particular taking
any v\in E and defining B to be the ball of radius \Vert v\Vert _B
centered at 0, the assumption (REF )
amounts to saying that \Vert v+ L(t,v)\Vert _... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.03630524501204491,
0.021295009180903435,
-0.044054433703422546,
0.024834008887410164,
0.0003260607772972435,
0.022347556427121162,
0.0012832684442400932,
0.027884868904948235,
0.05555617809295654,
-0.012241579592227936,
0.004698325879871845,
0.059522297233343124,
-0.0023892053868621588,
... |
be2d8d068c136cc8bfd926ac156194218ecc4175 | subsection | 40 | 84 | High-order method | The
algorithm that we are going to develop in §
relies on the construction of the low-order invariant domain
preserving solution _i described in
§REF -§REF and a high-order
solution _i that possibly wanders outside the invariant
domain. We are then going to limit the high-order solution by pushing
it back into the inva... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.04371355473995209,
0.012958383187651634,
0.0010388456284999847,
0.014667851850390434,
-0.007929187268018723,
0.04035567119717598,
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0.021154675632715225,
0.007307215593755245,
0.02538255788385868,
-0.05506931245326996,
0.006383797153830528,
-0.023016775026917458,
0.... |
f4a47003fc876a11b8a534860dda7c7c85b3d35e | subsection | 41 | 84 | Achieving high-order consistency | In this section we describe in broad terms how high-order consistency can be achieved. | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.017993951216340065,
0.0031153648160398006,
-0.04349682852625847,
0.02995939366519451,
-0.014666825532913208,
0.006024692207574844,
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0.026632267981767654,
0.013178776018321514,
0.03772777318954468,
-0.008737520314753056,
0.01180519163608551,
0.004429808352142572,
0.0... |
a0499be99bb2a7a6cf134ec6239cd98d127f83bc | subsection | 42 | 84 | Discretization-independent setting | Independently of the space discretization that is used, we henceforth assume that
the high-order update _i is computed as follows:\frac{m_i}{}(_i- _i^{n}) + \sum _{j\in (i)} _{ij}=m_i (_i^n),where the high-order flux _{ij} is assumed to be
skew-symmetric; _{ij}=-_{ji} for all
i\in , j\in (i) (under appropriate boundary... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.06909213215112686,
-0.026199467480182648,
-0.0243378859013319,
-0.013382022269070148,
-0.01035313867032528,
0.027557505294680595,
-0.0496523380279541,
0.0067596761509776115,
0.052276864647865295,
0.02029428817331791,
-0.03451554849743843,
-0.02001962810754776,
-0.037475768476724625,
0.0... |
6ea0328c89cceee3c906a9597d35919164e8a17f | subsection | 43 | 84 | High-order algebraic fluxes: Finite Volumes | In
the context of finite volume schemes, high-order algebraic fluxes
_{ij} are obtained as integrals of high-order numerical fluxes
over the interfaces between volumes, _{ij}:= \int _{\Gamma _{ij}} \widehat{}_{ij} s where
\widehat{}_{ij} is some numerical flux.
For instance, a widely popular choice of algebraic flux co... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.01987609826028347,
0.027350610122084618,
-0.02863195538520813,
-0.011799050495028496,
-0.019677795469760895,
0.012607517652213573,
0.005518172401934862,
0.0027114171534776688,
0.009747372940182686,
0.04820297285914421,
-0.03374208137392998,
0.02004389278590679,
-0.033467505127191544,
0.... |
3681da61a0be14d3b86b0003ecc3d59b152fe4a8 | subsection | 44 | 84 | High-order algebraic flux: Continuous Finite Elements | We now turn our attention to continuous finite elements. In
this case high-order consistency can be achieved by using a degenerate
graph viscosity d_{ij} such that d_{ij}\ll d_{ij} in smooth
regions while d_{ij}\approx d_{ij} near shocks. Of course
d_{ij} must also satisfy the conservation constraintsd_{ij}= d_{ji}\ge ... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.0394514836370945,
-0.019191788509488106,
-0.040031202137470245,
-0.02022918313741684,
-0.02280741184949875,
0.017834022641181946,
-0.0011089055333286524,
-0.005526411347091198,
0.00857375655323267,
0.05824661999940872,
-0.03813948482275009,
-0.02132759988307953,
-0.0077728270553052425,
... |
fe02dfe4ac46f0ac51f0f02ddfae32a2621f3421 | subsection | 45 | 84 | High-order algebraic flux: Continuous Finite Elements | \end{aligned}Then (REF ) holds with the following definition
for the high-order algebraic flux:_{ij}:=&{} \frac{m_{ij}}{}(_j-_j^n
- _i+ _i^n ) \\
& + ((_j^n) + (_i^n) ) _{ij} -
d_{ij}(^n_j - ^n_i).In the context of finite difference methods, a scheme with the above structure
is said to be linearly implicit as the numer... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.04034770652651787,
-0.01394773181527853,
-0.00253126909956336,
-0.019334547221660614,
-0.0061154854483902454,
0.007122652139514685,
0.030291300266981125,
0.02026541344821453,
0.01744229532778263,
0.02591165155172348,
-0.0169539712369442,
-0.02171512320637703,
-0.03573915734887123,
0.053... |
04abd6c247ed95e580b94b98fa9d45dcd0c0530b | subsection | 46 | 84 | High-order algebraic flux: Discontinuous Finite Elements | Just like for continuous finite
elements, high-order consistency is space is obtained for
discontinuous finite elements by replacing the low-order graph
viscosity d_{ij} by a high-order graph viscosity
d_{ij} satisfying the symmetry and positivity properties
stated in (REF ). The corresponding flux in
(REF ) is_{ij}:=
... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.03775535151362419,
-0.018297763541340828,
-0.04794960469007492,
-0.02649284154176712,
-0.01236129179596901,
-0.013536377809941769,
-0.000721075339242816,
-0.002596252830699086,
0.007164970971643925,
0.05173429474234581,
-0.010667337104678154,
-0.01913711056113243,
0.006730036344379187,
... |
81baeb1a4f4bb2ad6c502e3b63c1947751f3133b | subsection | 47 | 84 | Smoothness-based graph viscosity | The objective of this section is to present a method where the
high-order graph viscosity in (REF ),
(REF ), (REF ), or
(REF ) is obtained by estimating the smoothness
of some functional (an entropy) of the current solution. | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.0368136465549469,
-0.039866190403699875,
-0.012828304432332516,
-0.007928975857794285,
-0.01633109524846077,
0.0032624034211039543,
0.00905841588973999,
-0.0018057689303532243,
0.02945702336728573,
0.0146293044090271,
0.002142502460628748,
0.007574118208140135,
0.03293691948056221,
0.00... |
cb4a15eba62e2b65a05085df11de15f5977704c8 | subsection | 48 | 84 | Principles of the method | Let _h^n = \sum _{i\in } _i^n \varphi _i be the current
approximation and let g:\rightarrow be some functional (examples
will be given below). We define the
smoothness indicator associated to g as follows:\alpha _i^n:= \frac{\left|\sum _{j\in (i)}
\beta _{ij}(g(_j^n)-g(_i^n))\right|}{\max (\sum _{j\in (i)}
|\beta _{ij}... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.03881245106458664,
-0.005580052733421326,
-0.04061271250247955,
-0.030497675761580467,
0.01766698807477951,
-0.04500657692551613,
0.019360456615686417,
0.007273520343005657,
0.02563086338341236,
0.030131520703434944,
-0.026653047651052475,
0.021603155881166458,
-0.010214203968644142,
0.... |
4119af2cde191573beaa96987ddc3ce444481b87 | subsection | 49 | 84 | Principles of the method | For discontinuous elements, one could
take
\beta _{ij} = \int _{K}\varphi _j\varphi _ix -
\int _{\partial K} \frac{1}{2} \varphi _j _K \varphi _i
x,
where K is the unique cell such that i\in (K) and _K
is the unit normal vector on \partial K pointing outward K. For
finite volumes, one should get \alpha _i^n=0 if a line... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.06037917360663414,
0.012141464278101921,
-0.04102608934044838,
-0.02078777365386486,
-0.011622533202171326,
-0.04117871820926666,
0.05708243325352669,
0.015110055916011333,
-0.005685349460691214,
0.049603719264268875,
-0.044292304664850235,
0.0024897251278162003,
-0.015980029478669167,
... |
edde94e44e277ab6234ca98bbf5e028a044ecf50 | subsection | 50 | 84 | Stability | We now establish some invariant domain preserving properties
associated with the smoothness-based
graph viscosity (REF ) when the coefficients \beta _{ij} are positive.
We further specialize the setting
by assuming that g:\rightarrow is a projection onto one of the
scalar components of . Without loss of generality we s... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.04833467677235603,
-0.00348626053892076,
-0.021665163338184357,
-0.004191903863102198,
0.037929292768239975,
-0.0016048622783273458,
0.02764596976339817,
0.0453137569129467,
0.021405791863799095,
0.03329111635684967,
-0.029507342725992203,
0.027539169415831566,
-0.024594537913799286,
-0... |
07c5cb9aff6ca6329cb198e376cff229ef8a254a | subsection | 51 | 84 | Stability | Let _i^{n+1} be the high-order update
given by (REF ) using either the high-order cG
flux (REF ) or the high-order dG
flux (REF ) with any graph viscosity \lbrace d_{ij}\rbrace _{j\in (i)\backslash \lbrace i\rbrace } defined
by d_{ij}:= d_{ij}\max (\psi _i^n,\psi _j^n) with \psi _i^n,\psi _j^n\in [0,1]. Then,_i^{n+1}
&... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.05943424999713898,
-0.016774073243141174,
-0.033395517617464066,
-0.03415866941213608,
-0.00494522275403142,
0.01376725547015667,
-0.020070888102054596,
0.004239307250827551,
0.007616253569722176,
0.028862396255135536,
-0.01732354238629341,
-0.0033158939331769943,
-0.03205236792564392,
... |
1e9c13b7d40e15838840748212bb55320fdb4e76 | subsection | 52 | 84 | Stability | Then setting
_i^{*,n} := \frac{1}{\gamma _i^n}\sum _{j\in (i){\setminus }\lbrace i\rbrace }\frac{2d_{ij}}{m_i}\overline{}_{ij}^{n},
we have
_i= (1-\gamma _i^n)_i^n + \gamma _i^n _i^{*,n},
and this in turn implies that_i& = (1-\gamma _i^n)_i^n + \gamma _i^n _i^{*,n}
+ \frac{}{m_i}\sum _{j\in (i){\setminus }\lbrace i\rbr... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
0.001520685269497335,
0.02606998011469841,
-0.05006534233689308,
-0.03740409016609192,
-0.018778318539261818,
0.028114086017012596,
-0.033804021775722504,
0.04091263189911842,
0.0017294815042987466,
0.042651645839214325,
-0.020852932706475258,
0.028953084722161293,
-0.0528111569583416,
0.0... |
f0bcadd8228364d7c977ef20a3ec2c7ed03849e7 | subsection | 53 | 84 | Stability | With these definitions, the above inequality is rewritten as follows:_i&\le _i^{,n} + (_i^{,n} -_i^{,n}) \bigg ((1-\theta _i^n)(1-\gamma _i^n)
- \theta _i^n\frac{}{m_i}\sum _{j\in (i^-)} (d_{ij}-
d_{ij})\bigg ).(iii)
Using that d_{ij}\ge d_{ij}\psi _i^n and \psi _i^n\ge 0,
we infer that -d_{ij}\le -d_{ij}\psi _i^n, whi... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.017429325729608536,
0.031378891319036484,
-0.05414995551109314,
-0.03864365443587303,
0.0002184389013564214,
0.012278364971280098,
-0.001654984662309289,
0.032050423324108124,
0.004914398305118084,
0.03977305069565773,
-0.03632381558418274,
0.00412076897919178,
-0.029898466542363167,
0.... |
b5d57224e70ef9e483e503c1f960df6347f8b1d4 | subsection | 54 | 84 | Stability | Let i\in and n\ge 0.
Then, under the local CFL condition
\gamma _i^n \le \frac{1}{1+ k_\psi c_\sharp }, where
c_\sharp = \varpi ^\sharp \max _{i\in }\textup {card}((i))
(this number is uniformly bounded with respect to the mesh
sequence), the scheme is locally invariant domain preserving for the scalar component : _i\i... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.018935484811663628,
0.020674925297498703,
-0.054166775196790695,
-0.008247082121670246,
0.025038782507181168,
-0.02052234299480915,
0.022154973819851875,
0.03518551588058472,
0.029158510267734528,
0.02720545418560505,
0.009521146304905415,
0.05337334796786308,
-0.048765357583761215,
0.0... |
8731ed794a306022e49b7326c17fe8f844f50435 | subsection | 55 | 84 | Stability | This inequality in
turn implies that1-\alpha _i^n
&= 1- \frac{\left|\sum _{j\in (i^+)}
\beta _{ij}|_j^n-_i^n| - \!\sum _{j\in (i^-)}
\beta _{ij}|_j^n-_i^n|\right|}{\sum _{j\in (i)} \beta _{ij}|_j^n-_i^n|}
\\
&\le \frac{\sum _{j\in (i)} \beta _{ij}|_j^n-_i^n| + \sum _{j\in (i^+)}
\beta _{ij}|_j^n-_i^n| - \sum _{j\in (i^... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.008209466934204102,
0.027848100289702415,
-0.0830102264881134,
-0.020477890968322754,
0.01583908312022686,
-0.020477890968322754,
0.010727240703999996,
0.034577421844005585,
0.0014505807776004076,
0.030167503282427788,
-0.047089993953704834,
0.024384254589676857,
-0.021851221099495888,
... |
2ce85e276ac1fc3a20af54c91151f773b9abdc87 | subsection | 56 | 84 | Stability | Similarly, provided again that \gamma _i^{n} \le \frac{1}{1+ k_\psi c_\sharp },
we have\theta _i^n(1-\gamma _i^n)
- (1-\theta _i^n)(1- \psi (\alpha _i^n))\tfrac{1}{2} \gamma _i^{+,n} &\ge \theta _i^n(1-\gamma _i^n)
- k_\psi c_\sharp \theta _i^n(1- \theta _i^n)\gamma _i^{n} \\
&\ge \theta _i^n(1-(1+ k_\psi c_\sharp (1-\... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.0361902080476284,
0.004218630958348513,
-0.05153900384902954,
-0.007994800806045532,
0.021253354847431183,
-0.03393213450908661,
0.03228434920310974,
0.02940072864294052,
0.027051111683249474,
0.024625208228826523,
-0.011290367692708969,
0.04906732961535454,
-0.022458678111433983,
0.003... |
0ca8c4e26d1d5b2e275e3e672fe1a77a54dbf6a1 | subsection | 57 | 84 | Greedy graph viscosity | We continue
with a technique entirely based on the observations made in
Lemma REF , irrespective of any smoothness
considerations.
As in §REF ,
we specialize the setting by assuming that there is one scalar component of ,
say , for which the source term is zero, S\equiv 0.Let i\in and n\ge 0. Let \theta _{n}^n,
\gamma ... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.05474631488323212,
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0.027205318212509155,
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0.04543883353471756,
-0.007926619611680508,
-0.0... |
d76d145a16f5d185fbc57ea726e0b4e743b4478c | subsection | 58 | 84 | Greedy graph viscosity | The definition of \psi _i^n
in (REF ) implies that
\psi _i^n \ge 1-2\frac{1-\gamma _i^n}{\gamma _i^{-,n}}\frac{1-\theta _i^n}{\theta _i^n},
which in turn gives
\theta _i^n(\psi _i^n-1) \frac{1}{2} \gamma _i^{-,n} +
(1-\gamma _i^n)(1-\theta _i^n)\ge 0. This is the condition in
Lemma REF that shows that
_i^{n+1}\le _i^{,... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.043994154781103134,
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0.00020915432833135128,
0.025215789675712585,
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0.02526155300438404,
0.003916608635336161,
0.... |
62957b00f9e8615a3e0b645abbfa2cbde1cc496f | subsection | 59 | 84 | Commutator-based graph viscosity | The objective of this section is to construct the high-order graph
viscosity so that the method is entropy consistent and close to be
invariant domain preserving. In other words, we do not want to rely on
the (yet to be explained) limiting process to enforce entropy
consistency. For instance one naive choice consists o... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.07410561293363571,
-0.034641627222299576,
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0.004498070105910301,
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0.018205929547548294,
0.04678908735513687,
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0.017946498468518257,
0.004208118189126253,
... |
edbcb99b783a3c5ed309ffe1b99ff0bf73683947 | subsection | 60 | 84 | Convex Limiting | In this section we develop a general limiting framework to preserve
convex invariant sets and (more generally) quasiconcave constraints. This
work is aligned with the ideas presented in
KhobalattePerthame1994, PerthameYouchun1994,
PerthameShu1996 in the context of finite volume methods.
We also refer the
reader to Zhan... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.03806217759847641,
0.010896709747612476,
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0.026173468679189682,
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-0.011949753388762474,
-0.01188107579946518,... |
b72fc428aec033f74ec0bf91e0ff53ea92c80c40 | subsection | 61 | 84 | Quasiconcavity | We have seen in § that the low-order solution
_i satisfies some “convex bounds” and, in
principle, we would like the high-order solution to satisfy these
“convex bounds” as well. But, before proceeding any further, we need
to define clearly what we mean by convex bounds. We also need to
give a precise statement about t... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.023339364677667618,
-0.006021708715707064,
-0.05046183988451958,
0.02379699982702732,
0.008084878325462341,
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0.04643465578556061,
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-0.012386643327772617,
0.0021623235661536455,
0... |
0354ab68391e224b2951be34e878f7193dcc075d | subsection | 62 | 84 | Quasiconcavity | As a result, \Phi is concave
since R() \Psi () and -\chi R() are both concave
and the sum of two concave functions is concave (this may not be the
case for the sum of quasiconcave functions). Notice also that
\min _{j\in } \Phi (_j)\ge 0 because R\ge 0 and
\min _{j\in } \Psi (_j)-\chi \ge 0. Hence\Phi \Big (\sum _{j\in... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.04353782534599304,
0.017405977472662926,
-0.0381985679268837,
0.013454927131533623,
-0.003912912216037512,
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0.0289235170930624,
0.025414861738681793,
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0.039052847772836685,
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0.004912116099148989,
-0.0007570303278043866,
-0... |
f503b033c42b3e3261fe6daf8ddc8b7e8a949d7c | subsection | 63 | 84 | Quasiconcavity | A direct
computation shows that the functional \varepsilon :\rightarrow
has a negative semi-definite Hessian for every equation of state,
thereby proving that \varepsilon is concave, hence quasiconcave.Let us now illustrate the use of Lemma REF with R()=\rho .[Specific internal energy] Let
:=\lbrace =(\rho ,,E)^\in ^m... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.011383375152945518,
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0.024033265188336372,
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0.000052632374718086794,
-0.0072633870877325535,... |
c51083a1271285d9181bc45380f5cc1cc4c5982a | subsection | 64 | 84 | Quasiconcavity | Let k\in be such that
\Psi (_k) := \min _{j\in } \Psi (_j). Then, for any
j\in , we have \Psi (_k) \le \Psi (_j), which
implies that L\circ \Psi (_k) \le L\circ \Psi (_j). Hence
L (\min _{j\in }\Psi (_j)) = L(\Psi (_k)) =
\min _{j\in } L(\Psi (_j)). In conclusion
L\circ \psi (\sum _{j\in } \lambda _j_j) \ge \min _{j\in... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.010399690829217434,
-0.023225722834467888,
... |
dc9f88d4548cd55cfd85726221fe7191141a1014 | subsection | 65 | 84 | Bounds | In this section we define the
bounds that we are going to use to limit the high-order solution.
The following result will play a key role in the rest of the paper,
since it tells us precisely what are the “convex bounds” that the
low-order solution produced by the GMS-GV scheme satisfies.[Natural bounds on the GMS-GV s... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.05347776040434837,
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0.0017341353232041001,
-0.025487402454018593... |
eb851564c7daace0dfbab87f3eea09dca1e0869e | subsection | 66 | 84 | Bounds | The second statement \Psi (_i) \ge \Psi _i^{\min } is a
local bound that can be viewed as a local “generalized minimum
principle.” This bound cannot be made uniform; it is local in time
and space, since \Psi _i^{\min } depends on i and n.[Relaxation]
The reader must be aware that in general the bound \Psi _i^{\min }
de... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.04394324868917465,
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0... |
2631efd8215af935d7bd843a2accbbc5756f7781 | subsection | 67 | 84 | Abstract Framework | In the sections §REF , §REF and
§REF we have seen that most high-order methods can
be written in the algebraic form\frac{m_i}{}(_i-_i^n)
+ \sum _{j\in (i)} _{ij}= m_i(_i^n),with _{ij}\in ^m satisfying the skew-symmetry
constraint _{ij}= -_{ij} for all
j\in (i) (whether we use the consistent mass matrix for the
discreti... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.0235120989382267,
-0.024061374366283417,
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0.012160377576947212,
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0.017561599612236023,
0.014578721486032009,
0.03963947668671608,
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0.0021131897810846567,
-0.032041147351264954,
... |
a585c836e49c1b8d827135341a8b62dc331a068b | subsection | 68 | 84 | Convex limiting | Without loss of generality, we consider a family of quasiconcave
functionals \lbrace \Psi _i \rbrace _{i \in }, \Psi _i: \rightarrow
where \subset ^m is a convex set and
\Psi _i(_i)\ge 0 for each i \in . Or goal is to
modify the high-order update so that the modified high-order update
satisfies the same quasiconcave c... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.0851031169295311,
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0.04435252398252487,
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-0.02417563460767269,
-0.036232929676771164,
0... |
cafe6d8084e3441386a3a9644dfc43e40ade5629 | subsection | 69 | 84 | Convex limiting | We rewrite (REF ) as follows:_i^{n+1} = \sum _{j \in (i)\backslash \lbrace i\rbrace } \lambda _j (_i+_{ij}_{ij}^n),
\qquad \text{with} \qquad _{ij}^n := \frac{1}{m_i \lambda _j} _{ij}^n,where \lbrace \lambda _j\rbrace _{j \in (i)\backslash \lbrace i\rbrace } is any set of
strictly positive convex coefficients (see Rema... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.07117822766304016,
0.03912971168756485,
-0.010247894562780857,
-0.011232241988182068,
0.006905692629516125,
0.04221247509121895,
0.005299451295286417,
0.021213065832853317,
0.015299351885914803,
0.05753472074866295,
-0.03668792173266411,
0.012384463101625443,
-0.014635489322245121,
0.04... |
b5b5723c32f5b8c78a29a2d09d9d047b6298fe37 | subsection | 70 | 84 | Convex limiting | \end{array}\right.}The following two
statements hold true: (i)
\Psi _i(_i+ _{ij}^n)\ge 0 for every
\in [0,^i_j]; (ii) Setting
_{ij} = \min (^i_j, ^j_i), we have
\Psi _i(_i+ \ell _{ij}_{ij}^n)\ge 0 and
_{ij}=_{ji}.
(i) First, if
\Psi _i(_i+ _{ij}^n)\ge 0 we observe that
\Psi _i(_i+ _{ij}^n)\ge 0 for any
\in [0,1] beca... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.0718965083360672,
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0.059659454971551895,
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0.006091825198382139,
-0.02575579471886158,
0.0... |
7e4dee136bac77c898fa21c32faec0636e3a0614 | subsection | 71 | 84 | Convex limiting | It might be interesting though to
explore this question further; for instance, other choices of
convex coefficients could help preserve some symmetries.[Multiple limiting]
In general we have to consider families of quasiconcave functionals
\lbrace \lbrace \Psi _i\rbrace _{i \in }\rbrace _{l\in },
\Psi _i^l: ^l \rightar... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.06175851449370384,
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0.03594443202018738,
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0.0003125208895653486,
-0.041833460330963135,
-0... |
f4366e0d20e8c00300b55fcb1196431316edeac4 | subsection | 72 | 84 | Convex limiting | Since is a convex invariant set for
(REF ), the CFL assumption together with
Theorem REF implies that
_i\in for all i\in . Then
Theorem REF can be applied because
\Psi _i^l()\ge 0. This theorem then implies that
\Psi ^l(_i^{n+1})\ge \Psi _i^{l,\min } for all ł\in .
Moreover, _i^n+2(_i^n) \in and
\overline{}_{ij}^n\in ,... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.028008505702018738,
-0.039236314594745636,
0.03... |
bdec1d52c5844d819f6304bcdb5d91960cd9f7ae | subsection | 73 | 84 | Implementation details | The objective of this section to give further details on the convex
limiting technique introduced above in order to help the reader to
implement it. | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.... |
fb996bc073d035cf4d317ac495bb749576946fd1 | subsection | 74 | 84 | Pseudocode of the limiting algorithm | Given a set of quasi-convex functionals \lbrace \Psi _i\rbrace _{i \in },
\Psi _i:\rightarrow , such that
\Psi _i(_i) \ge 0 with convex set ,
Algorithm REF enforces the quasi-concave
constraints \Psi _i(_i^{n+1}) \ge 0 for each i \in .
This pseudocode attempts to reflect as accurately as possible the way
convex limitin... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.04219254106283188,
0.012462511658668518,
-0.04423657804727554,
0.010403222404420376,
0.003434055019170046,
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0.03874513879418373,
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-0.00650582741945982,
-0.02164541557431221,
0.01... |
821f0d0d0651862ecfd8161fc30a075061223b2c | subsection | 75 | 84 | Transforming | As mentioned in the previous subsection, the line-search invoked in
line REF and line REF of
Algorithm REF could be computationally
expensive. However, it happens sometimes that the constraint of
interest \Psi _i()\ge 0 can be transformed into
\tilde{\Psi }_i()\ge 0 where \tilde{\Psi }_i is a quadratic
function, not ne... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.02244454436004162,
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0.04562147334218025,
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-0.010611951351165771,
-0.0025976775214076042... |
cdb80a9de9bc3481b7c12205ddbbaf287a0107d6 | subsection | 76 | 84 | Transforming | Let
_j^{i} := \min (\ell ^{\min },\ell ^{\max }), then
\Psi (+ )\ge 0 for all
\in [0, _j^{i}].
Let us first observe that
\tilde{\Psi }(+ ) = a\ell ^2 + b\ell +
c\ell =:g(\ell ) for all \ell \in [0,\ell ^{\max }]; hence,
\Psi (+ )\ge 0 iff g(\ell )\ge 0 for all
\ell \in [0,\ell ^{\max }]. If there is no positive root ... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.05598783493041992,
0.03569796681404114,
-0.018840592354536057,
0.021220456808805466,
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0.03392832353711128,
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0.021006880328059196,
-0.026941286399960518,
0.001... |
52cd646d21d1fea5441e52ed430cf20d15ed5b93 | subsection | 77 | 84 | Transforming | Note that \tilde{\Psi }_i(+)\ge 0
iff \Psi _i(+)\ge 0 provided \rho (+)\ge 0.
Hence before applying Lemma REF , one must
compute the limiter ^{\max }, which depends on
and , such that \rho (+)\ge 0 for all
\in [0,^{\max }].
This technique has been introduced in
GuermondQuezadaPopovKeesFarthing2018 in the
context of th... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.050254352390766144,
0.011320197023451328,
-0.03774416074156761,
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0.024699999019503593,
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0.044090792536735535,
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-0.03322828561067581,
0... |
5421ca2fc9cd550bce8f0896f6f4a9b22a562033 | subsection | 78 | 84 | Transforming | Observe also that that -\Psi ^{\min }R:\rightarrow is
concave if R:\rightarrow is convex and
\Psi ^{\min }\ge 0. Hence the second
assertion is just a consequence of the concavity of \Phi : \rightarrow .[Specific
entropy] Let us
illustrate the use of Lemma REF with
the compressible Euler equations. Assume to simplify th... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.059788644313812256,
0.030092701315879822,
-0.00833195447921753,
0.004543662071228027,
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0.007404910400509834,
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-0... |
9b92e12e5010c8eaaf24411bea262c3d7b182a59 | subsection | 79 | 84 | Line-search: The Newton-secant solver | Unless the function g(\ell ) := \Psi _i(+ \ell _{ij}^n) has a
special structure (say, linear or quadratic), the line-searches
invoked at lines REF and REF in
Algorithm REF require the use of an iterative
procedure. Without claiming originality, we now show how the
line-searches can be done by using the Newton-secant al... | {
"cite_spans": []
} | 10.1016/j.cma.2018.11.036 | 1807.02563 | Invariant domain preserving discretization-independent schemes and
convex limiting for hyperbolic systems | [
"Jean-Luc Guermond",
"Bojan Popov",
"Ignacio Tomas"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
-0.07512563467025757,
0.008307455107569695,
-0.0168742798268795,
-0.01830844022333622,
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0.002546017523854971,
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0.0... |
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