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37bcf52 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 | import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Flow.CompactPeriodizationDerivatives
import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.CompactRapidDifferential
namespace OAI
/-! Local periodization preserves the full residual equation. A pressure
projection, when supplied by the torus Poisson solver, is a separate algebraic
step and does not change the velocity or its trajectories. -/
noncomputable section
open Set
open scoped BigOperators
namespace RapidForcing.CompactEmbedding
theorem field_chartSupported {u : Field Space} (hu : Supported u)
(hneg : ∀ t, t < 0 → ∀ x, u t x = 0) : ChartSupported (field u) := by
intro t x hx
by_cases ht : 0 ≤ t
· have hu' : unembed x ∉ K := by
intro h
exact hx (by simpa only [embed_unembed] using embed_mem_chart h)
simp [field, hu.zero_off ht hu']
· simp [field, hneg t (lt_of_not_ge ht)]
theorem periodic_divergence {u : Field Space} (hu : ChartSupported u) (t : ℝ) (x : Space) :
divergence (periodic u) t x = divergence u t (representative x) := by
unfold divergence
simp_rw [periodic_spatialD hu]
rfl
theorem periodic_advection {u : Field Space} (hu : ChartSupported u) (t : ℝ) (x : Space) :
advection (periodic u) t x = advection u t (representative x) := by
unfold advection
simp_rw [periodic_spatialD hu]
rfl
theorem periodic_laplace {u : Field Space} (hu : ChartSupported u) (t : ℝ) (x : Space) :
laplace (periodic u) t x = laplace u t (representative x) := by
unfold laplace
simp_rw [periodic_spatialD hu]
apply Finset.sum_congr rfl
intro i _
rw [periodic_spatialD (hu.spatialD i)]
rfl
theorem periodic_residual {u : Field Space} (hu : ChartSupported u) (ν t : ℝ) (x : Space) :
residual ν (periodic u) t x = residual ν u t (representative x) := by
unfold residual
rw [periodic_timeD, periodic_advection hu, periodic_laplace hu]
rfl
theorem periodic_navierStokes {u f : Field Space} {ν : ℝ}
(hu : ChartSupported u) (hNS : NavierStokes ν f u (fun _ _ => 0)) :
NavierStokes ν (periodic f) (periodic u) (fun _ _ => 0) := by
refine ⟨?_, ?_, ?_⟩
· intro t ht x
have h := hNS.1 t ht (representative x)
simp only [gradient, spatialD, fderiv_const_apply, zero_apply,
zero_smul, Finset.sum_const_zero, neg_zero, zero_add] at h ⊢
rw [periodic_timeD, periodic_advection hu, periodic_laplace hu]
exact h
· intro t ht x
rw [periodic_divergence hu]
exact hNS.2.1 t ht (representative x)
· intro x
exact hNS.2.2 (representative x)
theorem navierStokes_pressure_projection {u g : Field Space} {ν : ℝ}
{φ : Field ℝ} (hNS : NavierStokes ν g u (fun _ _ => 0)) :
NavierStokes ν (fun t x => g t x - gradient φ t x) u (fun t x => -φ t x) := by
refine ⟨?_, hNS.2⟩
intro t ht x
have h := hNS.1 t ht x
simp only [gradient, spatialD, fderiv_const_apply, zero_apply,
zero_smul, Finset.sum_const_zero, neg_zero, zero_add] at h
have hg : gradient (fun t x => -φ t x) t x = -gradient φ t x := by
unfold gradient spatialD
change (∑ i, fderiv ℝ (-(φ t)) x (basis i) • basis i) = _
rw [fderiv_neg]
simp only [neg_apply, neg_smul, Finset.sum_neg_distrib]
rw [hg, neg_neg, h]
module
end RapidForcing.CompactEmbedding
end
end OAI
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