text stringlengths 16 3.88k | source stringlengths 60 201 |
|---|---|
∂x2
+ ν
+ ν
Then a miracle occurs, the Blassius solution for δ � x is a graph of ux/U∞ vs. β = y U∞/νx; hits 0.99 at
ordinate of 5:
�
�
δ = 5.0
νx
.
U∞
Why 5.0, not 3.6? Because there must be vertical velocity due to mass conservation (show using differential
mass equation and integral box), carries lowxve... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6e459228e4e97c9c29e50c481b262c5c_lectures.pdf |
), diverges at x = 0! But δ � x does not hold there.
�
Now set to a friction factor:
τyx = −0.332
3
ρµU∞
x
= fx ·
2ρU∞
1
2
This time τ is not constant, so we have different fx = τ /K and fL = Fd/KA. Let’s evaluate both:
fx = 0.664
�
µ
ρU∞x
=
0.664
√
Rex
Also, note dimensionless BL thickness:
δ
x
�
νx
U∞
δ... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6e459228e4e97c9c29e50c481b262c5c_lectures.pdf |
channel flow between two parallel plates spaced apart a distance H, we can define
the entrance length Le as the point where the boundary layers from each side meet in the middle. The twin
Blassius functions are close enough to the parabolic profile that we can say it’s fullydeveloped at that point.
So we can plug in t... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6e459228e4e97c9c29e50c481b262c5c_lectures.pdf |
about “Rex ” and “ReL”? No significance, just different lengthscales, one for local and one for
global/average.
• Difference between uav and U ? Should be no uav for BL problems, sorry if I made a writo. D’oh!
∞
This is wrong, see next lecture’s notes.
• What’s up with δ/x? Just a ratio, dimensionless for convenience,... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6e459228e4e97c9c29e50c481b262c5c_lectures.pdf |
fficulty was wittily expressed in 1932 by the British physicist Horace Lamb, who,
in an address to the British Association for the Advancement of Science, reportedly said, “I am
an old man now, and when I die and go to heaven there are two matters on which I hope for
enlightenment. One is quantum electrodynamics, and ... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6e459228e4e97c9c29e50c481b262c5c_lectures.pdf |
of timesteps required
to approach steadystate will be prohibitively costly for many years, perhaps until long after Moore’s law
has been laid to rest (indeed, the roughly four petabytes of memory required to just store a single timestep
would cost about two billion dollars at the time this is being written). Furthe... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6e459228e4e97c9c29e50c481b262c5c_lectures.pdf |
eddy lengthscale as
� ∼ d √
1
Red
�
4
µ
.
µt
Even using the conservative estimate of µt = 30µ gives � ∼ 0.1 mm, in 1 meter cubed this gives a trillion
grid points, but you want a few grid points across each smallest eddy which means about a few3 � 100 times
more grid points, hence “tens of trillions”.
Put sli... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6e459228e4e97c9c29e50c481b262c5c_lectures.pdf |
100
H
.
So a large Reynolds number means a long entrance length (1000 means ten times the channel width),
and vice versa.
Energy cascade and the Kolmogorov microscale. Largest eddy Re=U L/ν, smallest eddy Reynolds number
u�/ν ∼ 1. Energy dissipation, W/m3; in smallest eddies:
� �2
du
dx
� = η
∼ η
2
u
�2
Assu... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6e459228e4e97c9c29e50c481b262c5c_lectures.pdf |
L =
1.328
√
,
ReL
In range 105 to 107, transition, oscillatory; beyond 107 fully turbulent. Always retains a laminar sublayer
against the wall, though it oscillates as vortices spiral down into it. New behavior:
δ
0.37
x Re0.2
x
=
So δ ∼ x0.8 . Grows much faster. Why? Mixing of momentum, higher effective velo... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6e459228e4e97c9c29e50c481b262c5c_lectures.pdf |
�
x =
∂t
∂u¯x
∂t
+ 0.
Same with spatial derivatives, pressure terms. But one thing which doesn’t timesmooth out:
This forms the Reynolds stresses, which we shift to the right side of the equation:
u
�
x
∂u�
∂x
x = 0.
τxy = −µ
�
�
−
¯
ρu� uyx
�
+
∂uy
∂x
∂ux
∂y
Show how it’s zero in the center of chann... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6e459228e4e97c9c29e50c481b262c5c_lectures.pdf |
1 for
heat and mass transfer.
Next time: thermal and solutal boundary layers, heat and mass transfer coefficients, turbulent boundary
layer, then natural convection. Last, Bernoulli equation, continuous reactors.
Modeling: K − � and K − � modeling (Cµ, C1, C2, σK and σ� are empirical constants):
K =
1
2
�
�
�
2 ... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6e459228e4e97c9c29e50c481b262c5c_lectures.pdf |
gives mixing time, and the
timescale of formation and elimination of these little eddies �2/ν. When attempting direct numerical
simulation of turbulence, this tells how small a timestep one will need (actually, a fraction of this for
accuracy); this also describes how long the eddies will last after the mixing power... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6e459228e4e97c9c29e50c481b262c5c_lectures.pdf |
blood platelets diffuse at around D = 10−9, but tumbling blood cells not only stir and increase
diffusivity, but somehow platelets end up on the sides of blood vessels, where they’re needed. I don’t fully
understand...
Heat and Mass Transfer Coefficients What about h? Start with hx, then hL, as before with fx and
fL. L... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6e459228e4e97c9c29e50c481b262c5c_lectures.pdf |
x, so graph T vs. x and vs. y, show yderiv is larger.
•
Is there a physical meaning behind δT /δu ∝ Pr−1/2 and δC /δu ∝ Pr−1/3? Yes, see below.
Heat and mass transfer coefficients Recap last time:
• Flow and heat/mass transfer: weakly coupled. So far, all laminar.
• Case 1: much larger thermal(/concentration) bound... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6e459228e4e97c9c29e50c481b262c5c_lectures.pdf |
1
� L
W L x=0
2(Ts − T∞) ��
L
hx(Ts − T∞)W dx = hL(Ts − T∞)
�L
kρcpU∞x/π
�
= hL(Ts − T∞)
x=0
hL = 2
kρcpU∞ = 2hx|x=L
πL
Now for case 2 (highPrandtl), need different formulation. Dimensional analysis of mass transfer:
Five parameters, two base units (cm, s), so three dimensionless. Eliminate x and D. Then one ... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6e459228e4e97c9c29e50c481b262c5c_lectures.pdf |
128Re1/2Pr1/2
x
1 + 0.90
Pr
√
High: (>0.6): nice derivation in W3R chapter 19:
Nux = 0.332Re1/2
Pr0.343
x
NuL = 0.664Re1/2
L
Pr0.343
Just as there are more correlations for f (friction factor), lots more correlations for various geometries etc.
in handout by 2001 TA Adam Nolte. Summarize: flow gives Re, props g... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6e459228e4e97c9c29e50c481b262c5c_lectures.pdf |
ρ
D�u
Dt
=
−�p + η�
2�u + ρ�g
DT
Dt
= α�2T +
q˙
ρcp
Full coupling comes in the ρ in the fluid flow equations.
Volumetric thermal expansion coefficient:
Note relation to 3.11 thermal expansion coeff:
β =
−
1 dρ
ρ dT
, ρ − ρ0 = β(T − T0)
α =
1 dL
L dT
β = −
1 dρ
ρ dT
= −
V d(M/V )
M
dT
d(1/V ) = −dV /V 2 ... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6e459228e4e97c9c29e50c481b262c5c_lectures.pdf |
erence across BL to
drive flow.
70
5. Also with small density diff: Δρ/ρ = βΔT (ρ is roughly linear with T ).
6. No edge effects (zdirection).
With assumptions 1 and 2, get momentum equation:
∂�u
∂t
+ �
u · ��
u = ν
∞�2�
u +
1
ρ
∞
(ρ�g − �p) .
Now for xmomentum, steadystate (assumption 3), assumption 4 give... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6e459228e4e97c9c29e50c481b262c5c_lectures.pdf |
+ gβ(T − T∞)
New dimensional analysis:
h = f (x, ν, k, ρcp, gβ, Ts − T∞)
Seven params 4 base units (kg, m, s, K); 3 dimless params. Again Pr (dim’less ρcp), Nu (dim’less h), this
time Grashof number (dim’less β).
Gr =
gβ(Ts − T
ν2
∞)L3
Forced convection: Nu = f (Re, Pr).
Natural convection: Nu = f (Gr, Pr).
Detour:... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6e459228e4e97c9c29e50c481b262c5c_lectures.pdf |
f (Pr)
x
�
4
Grx
4
Grx
4
Note velocity squared proportional to driving force in pipe flow, kinda same here; heat trans proportional
Grx for velocity, Nux ∝
to square root of velocity. Hence Rex ∝
Grx.
√
√
√
4
Transition to turbulence determined by Ra=GrPr, boundary at 109 Laminar, Ra between 104 and 109:
Rex ∝
.
�... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6e459228e4e97c9c29e50c481b262c5c_lectures.pdf |
upward for hot wall,
downward for cold. What’s the difference between velocity in the BL, far from it? Far from it, velocity
is zero.
• Why δu ≥ δT ? Hot region lifts (or cold region sinks) fluid, so all of the hot/cold region (thermal BL)
will be moving (in the velocity BL). For large Pr, ν > α, so the momentum diffu... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6e459228e4e97c9c29e50c481b262c5c_lectures.pdf |
Solutal buoyancy too,
dissolving salt cube.
βC =
−
1 dρ
.
ρ dC
Special: nucleate boiling, film boiling, h vs. T with liquid coolant.
If time: BL on rotating disk: u ∝ r, so uniform BL. Pretty cool.
Now can calculate (estimate) heat/mass transfer coefficients for forced and natural convection, laminar
or turbulent.... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6e459228e4e97c9c29e50c481b262c5c_lectures.pdf |
giant fatigue
specimen...
Visualizing 2D flows, giving approximate regions of large and small velocity. DON’T CROSS THE
STREAMS!
Concept: flow separation, difference between jet and inlet. Breathing through nose. (D’oh! Forgot to
mention breathing through the nose.)
Decisions... Finish the term with the Bernoulli e... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6e459228e4e97c9c29e50c481b262c5c_lectures.pdf |
∂s
Steadystate, constant ρ:
Integrate along a streamline:
In other words:
This is the Bernoulli equation.
�
�
∂ 1 ρu2
s +
2
∂s
∂p
∂s
dz
− ρgz = 0
ds
1
2
ρV 2 + p + ρgz = constant
KE + P + P E = constant
Example 1: draining tub with a hole in the bottom. Set z = 0 at the bottom: PE=ρgh at top, P
at ... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6e459228e4e97c9c29e50c481b262c5c_lectures.pdf |
ago (that was the “derive and solve a new equation”
problem of 2000), tendency for diff eqs and thought problems...
76
Semester summary You’ve come a very long way! Mentioned linear to multiple nonlinear PDEs, un
derstanding of solution. More generally, learned to start with a simple conservation relation: accum = ... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6e459228e4e97c9c29e50c481b262c5c_lectures.pdf |
P
patm
patm + ρgh1
ρg(h1 + h2) patm − ρgh2
ρg(h1 + h2)
patm
PE
ρg(h1 + h2)
ρgh2
ρgh2
0
Batch and Continuous Flow Reactors For those interested.
Basic definitions, motivating examples. Economics: batch better for flexibility, continuous for quality
and no setup time (always on).
Two types: volumetric and sur... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6e459228e4e97c9c29e50c481b262c5c_lectures.pdf |
, given volume V , homogeneous with constant k.
•
Batch:
prodection rate is
• Plug:
tR =
1
k
ln(CA,in/CA,out) =
4.6
k
V
4.6
k + tchange
=
kV
4.6 + ktchange
Q =
kV
ln(CA,in/CA,out)
=
kV
4.6
Better than batch, likely better quality too, less flexible.
• Perfect mixing:
Q =
kV
CA,in/CA,out − 1
=
kV ... | https://ocw.mit.edu/courses/3-185-transport-phenomena-in-materials-engineering-fall-2003/6e459228e4e97c9c29e50c481b262c5c_lectures.pdf |
3.032 Mechanical Behavior of Materials
Fall 2007
STRESS AND STRAIN TRANSFORMATIONS:
Finding stress on a material plane that differs from the one on which stress is known...
or ”Why it’s easier to remember Mohr’s circle”
Note: Derived in class on Wednesday 09.19.07.
Force balance for stress over a face inclined an ... | https://ocw.mit.edu/courses/3-032-mechanical-behavior-of-materials-fall-2007/6e90475e58ce6975b034dab5f48cc2a2_lec7.pdf |
to obtain the orientation
and
tan2θshearstress,max =
− (σxx−σyy)
2
τxy
τmax,in−plane =
�
(
σxx −
2
σyy )2 + τ2
xy
(6)
(7)
Note that the equations for coordinate transformations of strain (strain transformation equations)
are completely analogous. For example,
�x� x� =
�xx + �yy +
2
�xx − �yy cos2θ + �xy... | https://ocw.mit.edu/courses/3-032-mechanical-behavior-of-materials-fall-2007/6e90475e58ce6975b034dab5f48cc2a2_lec7.pdf |
6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007
Lecture 4-1
Lecture 4 - Carrier generation and
recombination
February 12, 2007
Contents:
1. G&R mechanisms
2. Thermal equilibrium: principle of detailed balance
3. G&R rates in thermal equilibrium
4. G&R rates outside thermal equilibrium
Reading ass... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/6ea95228c96b5cb849eca384b52f4ccf_lecture4.pdf |
recombination mechanisms
a) Band-to-band G&R, by means of:
• phonons (thermal G&R)
• photons (optical G&R)
Ec
Ev
heat
heat
hhυυ >> EgEg
hhυυ
thermal �
generation
thermal �
recombination
optical �
absorption
radiative �
recombination
• thermal G&R: very unlikely in Si, need too many phonons si
multaneou... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/6ea95228c96b5cb849eca384b52f4ccf_lecture4.pdf |
2007
Lecture 4-5
c) Trap-assisted generation and recombination, relying on elec
tronic states in middle of gap (”deep levels” or ”traps”) that arise
from:
• crystalline defects
• impurities
Ec
Et
Ev
trap-assisted�
thermal generation
trap-assisted�
thermal recombination
Trap-assisted G/R is:
•
dominant in S... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/6ea95228c96b5cb849eca384b52f4ccf_lecture4.pdf |
of detailed balance
Define:
Gi ≡ generation rate by process i [cm−3 s−1]
Ri ≡ recombination rate by process i [cm−3 s−1]
·
G ≡ total generation rate [cm−3 s−1]
R ≡ total recombination rate [cm−3 s−1]
·
·
·
In thermal equilibrium:
Ro = ΣRoi = Go = ΣGoi
Actually, detailed balance is also required:
Roi = Goi
for ... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/6ea95228c96b5cb849eca384b52f4ccf_lecture4.pdf |
depends of nopo:
R⇒
Ro,rad = rrad(T ) nopo
In TE, detailed balance implies:
2
grad = rradnopo = rradni
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YY... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/6ea95228c96b5cb849eca384b52f4ccf_lecture4.pdf |
of traps occupied by an electron:
nto = Ntf (Ei) = Nt ni + po
ni
Concentration of empty traps:
Nt − nto = Nt − Ntni + po
ni
po
= Ntni + po
Trap occupation depends on doping:
• n-type: po � ni → nto � Nt, most traps are full
• p-type: po � ni → nto � Nt, most traps are empty
Ec
Et
Ev
EF
Ec
Et
Ev
EF
n-ty... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/6ea95228c96b5cb849eca384b52f4ccf_lecture4.pdf |
ro,hc = ro,he
Then, relationships that tie up capture and emission coefficients:
ee = ceno
Nt − nto
nto
= ceni
eh = chpo
nto
Nt − nto
= chni
Capture coefficients can be calculated from first principles, but most
commonly they are measured.
Also define:
τeo =
1
Ntce
τho =
1
Ntch
τeo and τho are characteristi... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/6ea95228c96b5cb849eca384b52f4ccf_lecture4.pdf |
he =
ni
τeo
po
τho
If τeo not very different from τho,
ro,ec = ro,ee � ro,hc = ro,he
The rate at which trap communicates with CB much higher than
VB.
Ec
Et
Ev
EF
• lots of electrons in CB and trap ⇒ ro,ec = ro,ee high
• few holes in VB and trap ⇒ ro,hc = ro,he small
Reverse situation for p-type semiconduc... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/6ea95228c96b5cb849eca384b52f4ccf_lecture4.pdf |
concentrations change in time.
Useful to define net recombination rate, U:
U = R − G
Reflects imbalance between internal G&R mechanisms:
• if R > G → U > 0, net recombination prevails
• if R < G → U < 0, net generation prevails
• if R = G → U = 0, thermal equilibrium
If there are several mechanisms acting simultan... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/6ea95228c96b5cb849eca384b52f4ccf_lecture4.pdf |
np < nopo, Urad < 0, net generation prevails
• note: we have assumed that grad and rrad are unchanged from
equilibrium
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/6ea95228c96b5cb849eca384b52f4ccf_lecture4.pdf |
Auger R rate in TE is proportional to the square of the ma
jority carrier concentration and is linear on the minority carrier
concentration.
• Trap-assisted G/R rates in TE depend on the nature of the trap,
its concentration, the doping type and the doping level.
• In n-type semiconductor, midgap trap communicates... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/6ea95228c96b5cb849eca384b52f4ccf_lecture4.pdf |
Topic 1 Notes
Jeremy Orloff
1 Complex algebra and the complex plane
We will start with a review of the basic algebra and geometry of complex numbers. Most likely you
have encountered this previously in 18.03 or elsewhere.
1.1 Motivation
The equation 2 = −1 has no real solutions, yet we know that this equation arises... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/6eae09885b0fab3f068e31752198abea_MIT18_04S18_topic1.pdf |
of degree has exactly complex
roots (repeated roots are counted with multiplicity).
1Our motivation for using complex numbers is not the same as the historical motivation. Historically, mathematicians
were willing to say 2 = −1 had no solutions. The issue that pushed them to accept complex numbers had to do with
the... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/6eae09885b0fab3f068e31752198abea_MIT18_04S18_topic1.pdf |
442 = −44.
Before talking about division and absolute value we introduce a new operation called conjugation.
It will prove useful to have a name and symbol for this, since we will use it frequently.
Complex conjugation is denoted with a bar and defined by
+ = − .
If = + then its conjugate is = − and we read this as... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/6eae09885b0fab3f068e31752198abea_MIT18_04S18_topic1.pdf |
and to describe the complex number = + we can visualize
complex numbers as points in the -plane. When we do this we call it the complex plane. Since
is the real part of we call the -axis the real axis. Likewise, the -axis is the imaginary axis.
Imaginary axis
Imaginary axis
= + = (, )
= + = (, )
Real axis
−
... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/6eae09885b0fab3f068e31752198abea_MIT18_04S18_topic1.pdf |
:240)(cid:240) = magnitude = length = norm = absolute value = modulus
= arg() = argument of = polar angle of
As in 18.02 you should be able to visualize polar coordinates by thinking about the distance from
the origin and the angle with the -axis.
Example 1.5. Let’s make a table of , and for some complex numbers. ... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/6eae09885b0fab3f068e31752198abea_MIT18_04S18_topic1.pdf |
it’s a good definition.
To do that we need to show the e obeys all the rules we expect of an exponential. To do that
we go systematically through the properties of exponentials and check that they hold for complex
exponentials.
(1)
1.6.1 e behaves like a true exponential
P1. e differentiates as expected:
e
= e .
P... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/6eae09885b0fab3f068e31752198abea_MIT18_04S18_topic1.pdf |
2)!
∑
∞
+
(−1)
2+1
(2 + 1)!
0
= cos() + sin().
So the Euler formula definition is consistent with the usual power series for e.
Properties P1-P4 should convince you that e behaves like an exponential.
1.6.2 Complex exponentials and polar form
Now let’s turn to the relation between polar coordinates and comple... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/6eae09885b0fab3f068e31752198abea_MIT18_04S18_topic1.pdf |
e
−
.
In words: complex conjugation changes the sign of the argument.
Multiplication. If
1 and
1
2
2 then
2e
1e
=
=
2 =
1
1
2e(
1+
2).
1 COMPLEX ALGEBRA AND THE COMPLEX PLANE
7
This is what mathematicians call trivial to see, just write the multiplication down. In words, the
formula says the for
2 the ... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/6eae09885b0fab3f068e31752198abea_MIT18_04S18_topic1.pdf |
3
1.6.3 Complexification or complex replacement
In the next example we will illustrate the technique of complexification or complex replacement. This
can be used to simplify a trigonometric integral. It will come in handy when we need to compute
certain integrals.
Example 1.8. Use complex replacement to compute
Solu... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/6eae09885b0fab3f068e31752198abea_MIT18_04S18_topic1.pdf |
∫
= + = ∫
Clearly, by construction, Re( ) = as claimed above.
Alternative using polar coordinates to simplify the expression for :
In polar form, we have 1 + 2 = e, where = 5 and = arg(1 + 2) = tan−1(2) in the first
quadrant. Then:
ee2 .
ø
= ø
e(1+2)
5e
e
= ø
5
e(2−)
= ø (cos(2 − ) + sin(2 − )).
e
5
Th... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/6eae09885b0fab3f068e31752198abea_MIT18_04S18_topic1.pdf |
root when = 0, i.e. 21∕5e0. Likewise = 6 gives exactly the same root as = 1, and so on.
This means, we have 5 different roots corresponding to = 0, 1, 2, 3, 4.
= 21∕5, 21∕5e2∕5, 21∕5e4∕5, 21∕5e6∕5, 21∕5e8∕5
Similarly we can say that in general = e has distinct th roots:
= 1∕ e∕+ 2(∕) for = 0, 1, 2, … − 1.
Example 1... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/6eae09885b0fab3f068e31752198abea_MIT18_04S18_topic1.pdf |
the 5 fifth roots are
ø
21∕10e∕20, 21∕10e9∕20, 21∕10e17∕20, 21∕10e25∕20, 21∕10e33∕20.
Using a calculator we could write these numerically as + , but there is no easy simplification.
Example 1.13. We should check that our technique works as expected for a simple problem. Find
the 2 square roots of 4.
= 4e 2
Solution... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/6eae09885b0fab3f068e31752198abea_MIT18_04S18_topic1.pdf |
(cos() + sin()) = cos() + sin()
Proof. This is a simple consequence of Euler’s formula:
(cos() + sin())
= (e) = e
= cos() + sin().
The reason this simple fact has a name is that historically de Moivre stated it before Euler’s formula
was known. Without Euler’s formula there is not such a simple proof.
1 ... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/6eae09885b0fab3f068e31752198abea_MIT18_04S18_topic1.pdf |
cos − sin
cos − sin
cos
sin
cos
sin
0
0
=
] [
=
[
corresponding to a stretch factor multiplied by a 2D rotation matrix. In particular, multiplication
by corresponds to the rotation with angle = ∕2 and = 1.
We will not make a lot of use of the matrix representation of complex numbers, but later it will he... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/6eae09885b0fab3f068e31752198abea_MIT18_04S18_topic1.pdf |
this. If not: ask a teacher or TA.
8. The path e for 0 < < ∞ wraps counterclockwise around the unit circle. It does so infinitely
many times. This is illustrated in the following picture.
The map → e wraps the real axis around the unit circle.
1.11 Complex functions as mappings
A complex function = () is hard to gra... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/6eae09885b0fab3f068e31752198abea_MIT18_04S18_topic1.pdf |
Next, we’ll illustrate visualizing mappings with some examples:
Example 1.14. The mapping = 2. We visualize this by putting the -plane on the left and the
-plane on the right. We then draw various curves and regions in the -plane and the corresponding
image under 2 in the -plane.
In the first figure we show that rays... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/6eae09885b0fab3f068e31752198abea_MIT18_04S18_topic1.pdf |
2 = (−)2.
Re(z)Im(z)0.512340.51234Re(w)Im(w)124688101214168162432z7!w=z2Re(z)Im(z)0.51234(cid:0)1(cid:0)2(cid:0)3(cid:0)4Re(w)Im(w)124688101214168162432z7!w=z2 1 COMPLEX ALGEBRA AND THE COMPLEX PLANE
15
Vertical stripes in quadrant 4 are mapped identically to vertical stripes in quadrant 2.
Simplified view of the fir... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/6eae09885b0fab3f068e31752198abea_MIT18_04S18_topic1.pdf |
1+πi/2Re(w)Im(w)1(cid:2)e1e2(cid:2)z7!w=ezRe(z)Im(z)012(cid:0)1πi/22πiπi(cid:0)πiRe(w)Im(w)1e1e2z7!w=ezRe(z)Im(z)012(cid:0)1πi/22πiπi(cid:0)πiRe(w)Im(w)1e1e2z7!w=ez 1 COMPLEX ALGEBRA AND THE COMPLEX PLANE
17
Simplified view showing e maps the horizontal stripe 0 ≤ < 2 to the punctured plane.
Simplified view showing e ... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/6eae09885b0fab3f068e31752198abea_MIT18_04S18_topic1.pdf |
+2 has the same value for every integer .
Re(z)Im(z)0πi2πiRe(w)Im(w)z7!w=ezRe(z)Im(z)0πi(cid:0)πiRe(w)Im(w)z7!w=ez 1 COMPLEX ALGEBRA AND THE COMPLEX PLANE
18
1.12.2 Branches of arg()
Important note. You should master this section. Branches of arg() are the key that really underlies
all our other examples. Fortunat... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/6eae09885b0fab3f068e31752198abea_MIT18_04S18_topic1.pdf |
uity. If we need arg() to be continuous
we will need to remove (cut) the points of discontinuity out of the domain. The branch cut for this
branch of arg() is shown as a thick orange line in the figure. If we make the branch cut then the
domain for arg() is the plane minus the cut, i.e. we will only consider arg() fo... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/6eae09885b0fab3f068e31752198abea_MIT18_04S18_topic1.pdf |
(v) We won’t make use of this in 18.04, but, in fact, the branch cut doesn’t have to be a straight line.
Any curve that goes from the origin to infinity will do. The argument will be continuous except for
xyarg=0arg=π/4arg=π/2arg=3π/4arg=πarg=−3π/4arg=−π/2arg=−π/4arg≈0arg≈−πxyarg=2πarg=π/4arg=π/2arg=3π/4arg=πarg=5π/4ar... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/6eae09885b0fab3f068e31752198abea_MIT18_04S18_topic1.pdf |
(image) of is the set of all () for in the domain, i.e. the set of all reached
by .
Branch. For a multiple-valued function, a branch is a choice of range for the function. We choose
the range to exclude all but one possible value for each element of the domain.
Branch cut. A branch cut removes (cuts) points out of t... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/6eae09885b0fab3f068e31752198abea_MIT18_04S18_topic1.pdf |
(),
where log((cid:240)(cid:240)) is the usual natural logarithm of a positive real number.
Remarks.
1. Since arg() has infinitely many possible values, so does log().
2. log(0) is not defined. (Both because arg(0) is not defined and log((cid:240)0(cid:240)) is not defined.)
3. Choosing a branch for arg() makes log() ... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/6eae09885b0fab3f068e31752198abea_MIT18_04S18_topic1.pdf |
showing = log() as a mapping
The figures below show different aspects of the mapping given by log().
In the first figure we see that a point is mapped to (infinitely) many values of . In this case we
show log(1) (blue dots), log(4) (red dots), log() (blue cross), and log(4) (red cross). The values in
the principal branc... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/6eae09885b0fab3f068e31752198abea_MIT18_04S18_topic1.pdf |
a single horizontal line in the principal (shaded) region
of the -plane.
Mapping log(): mapping circles and rays
1.14.2 Complex powers
We can use the log function to define complex powers.
Definition. Let and be complex numbers then the power is defined as
= e log().
This is generally multiple-valued, so to specify... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/6eae09885b0fab3f068e31752198abea_MIT18_04S18_topic1.pdf |
5∕6,
e9∕6
On the principal branch log() =
, so the value of 1∕3 which comes from this is
2
e∕6
=
ø
3
2
+
.
2
Example 1.23. Compute all the values of 1. What is the value from the principal branch?
Solution: This is similar to the problems above. log(1) = 2, so
1 = e log(1) = e2
= e
−2
, where is an inte... | https://ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/6eae09885b0fab3f068e31752198abea_MIT18_04S18_topic1.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
18.917 Topics in Algebraic Topology: The Sullivan Conjecture
Fall 2007
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
The Injectivity of H ∗(BV ) (Lecture 9)
Let n be a nonnegative integer, and let SqI be an element of the S... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/6eb13ec5cf01a6487074443b2264771a_lecture9.pdf |
n)). We may
therefore reformulate Proposition 1 as follows:
Proposition 2. Let m and n be nonnegative integers. Then there is a canonical isomorphism
HomFun(Symn , Symm) � HomA(F (m), F (n)).
Let U denote the category of unstable modules over the Steenrod algebra. Unwinding the definitions,
we see that the isomorph... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/6eb13ec5cf01a6487074443b2264771a_lecture9.pdf |
(F, G) � HomFun(DG, DF ).
1
Example 4. For each n ≥ 0, we let Γn : Vectf → Vect denote the functor
V �→ (V ⊗n)Σn
.
Then Γn is isomorphic to the dual D Symn .
We can reformulate Proposition 2 as follows:
Proposition 5. Let m and n be nonnegative integers. Then there is a canonical isomorphism
HomFun(Γm , Γn) � H... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/6eb13ec5cf01a6487074443b2264771a_lecture9.pdf |
closed under the formation of subobjects,
quotient objects, and extensions in the category Fun.
Remark 8. Let F ∈ Fun be a functor which takes values in finite dimensional vector space, and let
dF : Z≥0 → Z≥0 be the function defined by the formula
dF (n) = dim F (Fn
2 ).
We note that dΔ(F )(n) = dF (n + 1) − dF (n)... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/6eb13ec5cf01a6487074443b2264771a_lecture9.pdf |
and the image of
F (V ) coincides with F (n)(V ). We can then define F (n) = Im(α). Then F (n) is a quotient
each map G(V )
of G, and therefore polynomial of degree ≤ n. It is easy to see that F (n) has the desired properties.
→
2
Definition 11. A functor F ∈ Fun is analytic if it is the union of the polynomial sub... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/6eb13ec5cf01a6487074443b2264771a_lecture9.pdf |
a sequence of polynomial subfunctors Fα
polynomial functors
⊕αF (n).α
We will need the following result, whose proof we defer until the next lecture:
Proposition 13. The category Funan of analytic functors is generated by the objects {Γn}n≥0.
Combining this with the results of the previous lecture, we obtain the f... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/6eb13ec5cf01a6487074443b2264771a_lecture9.pdf |
then let M n denote
the value of M on the object F (n) ∈ R. For every n and every Steenrod operation SqI , we have an object
SqI νn ∈ F (n), which we can identify with a map F (n + deg(I)) → F (n) in R. This determines a map
M n M n+deg(I).
→
It is easy to see that this endows M with the structure of a graded A-mod... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/6eb13ec5cf01a6487074443b2264771a_lecture9.pdf |
Fun(PV vee , DF ) � DF (V ∨) = F (V )∨.
This is evidently an exact functor of F , so that IV is an injective object of Fun. We observe that IV can be
described by the formula
W �→ FHom(W,V )
.
2
Proposition 16. Let V be a finite dimensional vector space over F2. Then the functor IV is analytic.
Proof. We observe ... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/6eb13ec5cf01a6487074443b2264771a_lecture9.pdf |
It is easy to identify this object: we have
(GIV )n = HomFun(Γn, IV ) � Γn(V )∨ = Symn(V ∨) � Hn(BV ).
It is not hard to show that this identification is compatible with the action of the Steenrod algebra. Conse
quently, we have proven the following:
Proposition 17. Let V be a finite dimensional vector space over F2.... | https://ocw.mit.edu/courses/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/6eb13ec5cf01a6487074443b2264771a_lecture9.pdf |
Gas exchange Processes
To move working fluid in and out of engine
• Engine performance is air limited
• Engines are usually optimized for maximum
power at high speed
Considerations
• 4-stroke engine: volumetric efficiency
• 2-stroke engine: scavenging/ trapping efficiency
• Charge motion control; tuning; noise ... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/6ec38b3fc5493e328115e0a49a565f35_MIT2_61S17_lec8.pdf |
HV, and lower
stoichiometric air/fuel ratio
– In practice, most heat from the wall unless
direct injection is used
Volumetric efficiency: quasi-static effects
(cont.)
• Air displacement by fuel and water vapor
(cid:1876)(cid:3556)(cid:3028)=
Dry air
Fig. 6.3
V is volume inducted
i
P is intake pressure
i
i ... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/6ec38b3fc5493e328115e0a49a565f35_MIT2_61S17_lec8.pdf |
loss
Throttle loss
Intake flow loss
Fig. 13-15
© McGraw-Hill Education. All rights reserved. This content is excluded from our Creative
Commons license. For more information, see https://ocw.mit.edu/help/faq-fair-use.
4
Volumetric Efficiency: dynamic effects
cont.
Ram effect
– Due to fluid in... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/6ec38b3fc5493e328115e0a49a565f35_MIT2_61S17_lec8.pdf |
��
RT1 1
m
P2
P1
m
P2/P1
1
1
P2
P1
critical
0.528 for 1.4; increases with
2
1
Volumetric Efficiency: dynamic effects
cont.
Overlap back flow
– Back flow of burned gas from
exhaust/cylinder to intake port
– Increases residual gas fraction
– Prom... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/6ec38b3fc5493e328115e0a49a565f35_MIT2_61S17_lec8.pdf |
see https://ocw.mit.edu/help/faq-fair-use.
2-Stroke engine gas exchange
Delivery ratio
Air mass delivered per cycle
V
a,0 D
Trapping efficiency
t
Air mass retained
Air mass delivered
Air mass retained
m
a
V
a,0 D
t
Scavenging ratio
sc
Ai
r mass retained
Trapped charge mas... | https://ocw.mit.edu/courses/2-61-internal-combustion-engines-spring-2017/6ec38b3fc5493e328115e0a49a565f35_MIT2_61S17_lec8.pdf |
6.087 Lecture 3 – January 13, 2010
Review
Blocks and Compound Statements
Control Flow
Conditional Statements
Loops
Functions
Modular Programming
Variable Scope
Static Variables
Register Variables
1
Review: Definitions
• Variable - name/reference to a stored value (usually in
memory)
• Data type - determine... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/6ef53f22595564c47c0c377ef8bd5398_MIT6_087IAP10_lec03.pdf |
temp2 = x∗y ;
z += bar ( temp2 ) ;
}
}
6
6.087 Lecture 3 – January 13, 2010
Review
Blocks and Compound Statements
Control Flow
Conditional Statements
Loops
Functions
Modular Programming
Variable Scope
Static Variables
Register Variables
7
Control conditions
• Unlike C++ or Java, no boolean type (in C89/C9... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/6ef53f22595564c47c0c377ef8bd5398_MIT6_087IAP10_lec03.pdf |
( x + 1 ) / 2 ;
• Additional alternative control paths
• Conditions evaluated in order until one is met; inner
statement then executed
• If multiple conditions true, only first executed
• Equivalent to nested if statements
11
Nesting if statements
i f ( x % 4 == 0 )
i f ( x % 2 == 0 )
y = 2 ;
else
y = 1 ;
T... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/6ef53f22595564c47c0c377ef8bd5398_MIT6_087IAP10_lec03.pdf |
:
/ ∗ do something
’Y ’ ∗ /
ch ==
i f
case ’N’ :
/ ∗ do something
’Y ’ o r
’N ’ ∗ /
ch ==
ch ==
i f
break ;
}
15
The switch statement
•
Contents of switch statement a block
• Case labels: different entry points into block
• Similar to labels used with goto keyword (next lecture. . . )
16
Loop stat... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/6ef53f22595564c47c0c377ef8bd5398_MIT6_087IAP10_lec03.pdf |
/ ∗
i n c r e m e n t ∗ /
j ∗= i ;
i ++; / ∗
}
r e t u r n
j ;
}
20
The for loop
• Compound expressions separated by commas
i n t
f a c t o r i a l ( i n t n ) {
i ,
i n t
f o r ( i = 1 ,
j ;
;
r e t u r n
j ;
}
j = 1 ;
i <= n ;
j ∗= i ,
i ++)
• Comma: operator with lowest precedence, evaluated
left... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/6ef53f22595564c47c0c377ef8bd5398_MIT6_087IAP10_lec03.pdf |
( / ∗ o t h e r c o n d i t i o n s ∗ / ) ;
23
The continue keyword
• Use to skip an iteration
• continue; skips rest of innermost loop body, jumping to loop
condition
• Example:
# define min ( a , b ) ( ( a ) < ( b ) ? ( a )
: ( b ) )
i n t gcd ( i n t a ,
i n t b ) {
i n t
f o r ( i = 2 ;
i , r e t = 1 , m... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/6ef53f22595564c47c0c377ef8bd5398_MIT6_087IAP10_lec03.pdf |
get a, b, c from command line
compute g = gcd(a,b)
if (c is not a multiple of the gcd)
no solutions exist; exit
run Extended Euclidean algorithm on a, b
rescale x and y output by (c/g)
print solution
• Extended Euclidean algorithm: finds integers x, y s.t.
ax + by = gcd(a, b).
26
Computing the gcd
• Compute the gcd... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/6ef53f22595564c47c0c377ef8bd5398_MIT6_087IAP10_lec03.pdf |
Extended Euclidean algorithm returns gcd, and two other
state variables, x and y
• Functions only return (up to) one value
• Solution: use global variables
• Declare variables for other outputs outside the function
• variables declared outside of a function block are globals
• persist throughout life of program
... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/6ef53f22595564c47c0c377ef8bd5398_MIT6_087IAP10_lec03.pdf |
The extern keyword
• Need to inform other source files about functions/global
variables in euclid.c
• For functions: put function prototypes in a header file
• For variables: re-declare the global variable using the
extern keyword in header file
• extern informs compiler that variable defined somewhere
else
• Enable... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/6ef53f22595564c47c0c377ef8bd5398_MIT6_087IAP10_lec03.pdf |
. ∗ /
g = e x t _ e u c l i d ( a , b ) ;
• Results in global variables x and y
/ ∗ r e s c a l e so ax+by = c ∗ /
grow = c / g ;
x ∗= grow ;
y ∗= grow ;
36
Compiling with the Euclid module
• Just compiling diophant.c is insufficient
• The functions gcd() and ext_euclid() are defined in
euclid.c; this source fi... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/6ef53f22595564c47c0c377ef8bd5398_MIT6_087IAP10_lec03.pdf |
, n ;
p r i n t f ( "%3d: %d\n" , 1 , a ) ;
p r i n t f ( "%3d: %d\n" , 2 , b ) ;
f o r ( n = 3 ; n <= nmax ; n++) {
c = a + b ; a = b ; b = c ;
p r i n t f ( "%3d: %d\n" , n , c ) ;
}
r e t u r n 0 ;
}
/ ∗ success ∗ /
39
Scope and nested declarations
How many lines are printed now?
i n t nmax = 2 0 ;
/ ∗... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/6ef53f22595564c47c0c377ef8bd5398_MIT6_087IAP10_lec03.pdf |
are initialized only during program initialization
• do not get reinitialized with each function call
s t a t i c
i n t somePersistentVar = 0 ;
41
Register variables
• During execution, data processed in registers
• Explicitly store commonly used data in registers – minimize
load/store overhead
• Can explicitl... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/6ef53f22595564c47c0c377ef8bd5398_MIT6_087IAP10_lec03.pdf |
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MIT OpenCourseWare
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6.087 Practical Programming in C
January (IAP) 2010
For information about citing these materials or our Terms of Use,visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/6ef53f22595564c47c0c377ef8bd5398_MIT6_087IAP10_lec03.pdf |
16.920J/SMA 5212
Numerical Methods for Partial Differential Equations
Lecture 5
Finite Differences: Parabolic Problems
B. C. Khoo
Thanks to Franklin Tan
19 February 2003
16.920J/SMA 5212 Numerical Methods for PDEs
Slide 2
OUTLINE
•
• ... | https://ocw.mit.edu/courses/16-920j-numerical-methods-for-partial-differential-equations-sma-5212-spring-2003/6f02243dbad2edddbca94cbe8bdce5f1_lec5_notes.pdf |
1
+
O x
(
2
)
which is second-order accurate.
•
Schemes of other orders of accuracy may be constructed.
Slide 4
Slide 5
Construction of Spatial Difference Scheme of Any Order p
The idea of constructing a spatial difference operator is to represent the spatial
differential operator at a location by the neighborin... | https://ocw.mit.edu/courses/16-920j-numerical-methods-for-partial-differential-equations-sma-5212-spring-2003/6f02243dbad2edddbca94cbe8bdce5f1_lec5_notes.pdf |
is determined at the end of the analysis
when the a ’s are made known.)
This column consists of all the terms on the
LHS of (1).
uj
0
a -
1
a
0
a
1
¢
uj¢
1
x a -
1
0
x a
1
uj¢
¢
0
1
2
1
2
2
x a -
1
0
2
x a
1
uj¢
¢
0
1
6
1
6
3
x a -
1
0
3
x a
1
ju ¢
1jua -
1
jua
0
1jua
+
1
¢ +
u
j
=
1
k
=-
k
1
ua
k
+
j... | https://ocw.mit.edu/courses/16-920j-numerical-methods-for-partial-differential-equations-sma-5212-spring-2003/6f02243dbad2edddbca94cbe8bdce5f1_lec5_notes.pdf |
S
3
1
2
= -
S
4
2
x
a
1
6
3
x
a
¢ +
u
j
=
1
k
=-
k
1
a
u
k
+
j k
=
S
1
+
S
2
+
S
3
+
S
4
+
....
3.
Make as many
iS ’s as possible vanish by choosing appropriate
a
’s.
k
In this instance, since we have three unknowns
therefore set:
a -
a
,
1
0
and
a
1
, we can
=
0
=
=
0
0
S
1
S
2
S
3
(Note that in the Taylor S... | https://ocw.mit.edu/courses/16-920j-numerical-methods-for-partial-differential-equations-sma-5212-spring-2003/6f02243dbad2edddbca94cbe8bdce5f1_lec5_notes.pdf |
(cid:215)
(cid:215)
(cid:8)
(cid:9)
¢
¢
D
(cid:215)
(cid:215)
(cid:10)
(cid:11)
(cid:12)
(cid:13)
(cid:8)
(cid:9)
¢
¢
¢
D
(cid:215)
(cid:215)
(cid:10)
(cid:11)
(cid:12)
(cid:13)
16.920J/SMA 5212 Numerical Methods for PDEs
4.
Substituting the
a
k
’s into
u
¢ +
j
=
1
k
=-
k
1
a
u
k
+
j k
=
S
1
+
S
2
+
S
3
+
S
4
+
..... | https://ocw.mit.edu/courses/16-920j-numerical-methods-for-partial-differential-equations-sma-5212-spring-2003/6f02243dbad2edddbca94cbe8bdce5f1_lec5_notes.pdf |
j
+
du
dx
j
u
4
2
j
+
1
x
+
u
+
j
2
=
(
O x
2
)
which is also second-order accurate.
(We can also use a similar procedure to construct the finite difference scheme
of Hermitian type for a spatial operator. This is not covered here).
6
(cid:0)
-
¢
¢
¢
¢
-
-
(cid:215)
D
-
-
(cid:1)
(cid:2)
D
(cid:3)
(... | https://ocw.mit.edu/courses/16-920j-numerical-methods-for-partial-differential-equations-sma-5212-spring-2003/6f02243dbad2edddbca94cbe8bdce5f1_lec5_notes.pdf |
:2)
(cid:3)
(cid:2)
(cid:3)
(cid:2)
(cid:3)
(cid:2)
(cid:3)
(cid:2)
(cid:3)
(cid:2)
(cid:3)
(cid:2)
(cid:3)
(cid:2)
(cid:3)
(cid:2)
(cid:3)
(cid:2)
(cid:3)
(cid:2)
(cid:3)
(cid:2)
(cid:3)
(cid:2)
(cid:3)
(cid:2)
(cid:3)
(cid:2)
(cid:3)
(cid:2)
(cid:3)
(cid:2)
(cid:3)
(cid:2)
(cid:3)
(cid:2)
(cid:2)
(cid:3)
(cid:2)
(cid... | https://ocw.mit.edu/courses/16-920j-numerical-methods-for-partial-differential-equations-sma-5212-spring-2003/6f02243dbad2edddbca94cbe8bdce5f1_lec5_notes.pdf |
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