lecture 3 - mathematicsmath.mit.edu/~stoopn/18.086/lecture3-4.pdf · lecture 3 18.086. info ......
TRANSCRIPT
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Lecture 318.086
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Info• PSet 1 will be posted this afternoon/evening (due in 2 weeks)
• Until Feb. 22: Decide for a computational project (50% grade) and submit proposal to me by email (content: see next slide)
• March 28: Submit short (max 1page) written mid-term project report
• Until Friday, May 6: Submit written final project report
• May 5, 10, 12: Presentation (in class): 12 minutes + 3 minutes questions/discussion
• Content of project: Free choice, as long as related to course contents/computational aspect(can also go beyond!). No recycling of existing/old projects!Implementation/testing/comparison of methods is ok
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Term project• When submitting your project proposal (Feb. 22), use this template:
• Project title
• Project background: Does it relate to your work in another field (e.g. your thesis)? If yes, briefly outline the questions and goals of your work in the other field.
• Questions and Goals: Briefly describe the questions you wish to investigate in your project. What are your expectations?
• Plan: Which language do you plan to program in? Do you intend to use special software? Does your project work relate to the work of other people at MIT?
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Plan for today• Runge-Kutta time integration
• Basic time integration algorithm
• 1-way wave equation
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FD schemes for ODEs• Equation: u’ = f(u) with u(0) given
• First order accuracy (p=1): Euler methods
• Second order (p=2): global error O(Δt2), local error O(Δt3):
Trapezoidal implicit
Adams-Bashford explicit
Backward differences explicit
Runge-Kutta 2 explicit
Un+1 � Un
�t=
1
2[f(Un+1, tn+1) + f(Un, tn)]
Un+1 � Un
�t=
3
2f(Un, tn)�
1
2f(Un�1, tn�1)
Un+1 � Un
�t=
1
2[f(Un, tn) + f(Un +�tf(Un, tn), tn+1)]
Lecture: Stability & Accuracy
6.2 Finite Difference Methods 467
3un+1 - 4 u n + un-1 Backward differences/BDF2 2At = f (Un+l, tn+l) . (19)
What about stability? The trapezoidal method (15) is stable even for stiff equa- tions, when a is very negative: l - $a At (left side) will be larger than 1 + l a At (right side). (19) is even more stable and accurate. The Adams method (17) will be stable if At is small enough, but there is always a limit on At for explicit systems.
Here is a quick way to find the stability limit -a& 5 C in (17) when a is real. The limit occurs when the growth factor is exactly G = -1. Set Un+l = - 1 and Un = 1 and Un-l = - 1 i n (17 ) . Solve for a when f (u , t ) = au:
-2 3 Stability limit in (17) - - - 1 - a + - a gives a n t = -1. So C = 1 . (20)
a t 2 2
We now have three second-order methods (15)-(17)-(19), all definitely useful. The reader might suggest including both Un-l and fn-1 to increase the accuracy to third order. Sadly, this method is violently unstable (Problem 5). We may extend (19) by older values of U in backward differences, or extend (17) by older f (U). But including both U and f (U) for extreme accuracy produces instability for all At.
Multistep Methods: Explicit and Implicit
By using p earlier values of U , the accuracy can be increased to order p. Backward Euler has p = 1, and BDF2 in (19) has p = 2. Each VU is U(t) - U(t - At):
1 1 Backward differences (V + Z ~ 2 + . . + -VP) Un+1 = At f (Un+1, tn+l) . (21) P
MATLAB's stiff code odel5s varies from p = 1 to p = 5 depending on the local error. The alternative is to use older values of f (U, t) instead of U. Explicit comes first:
Adams- Bashforth Un+1 - Un = At(b1 fn + . . . + bp fn-,+I) (22) The table shows the numbers b up to p = 4, starting with Euler for p = 1.
The fourth-order method is often a good choice, although astronomers go above p = 8.
order of accuracy p = l p = 2 p = 3 p = 4
b l b2 b3 b4
1 312 -112
23/12 -16112 5/12 55/24 -59124 37/24 -9124
limit on -aAt for stability
2 1
611 1 3/10
constant c in error DE
5/ 12 112
318 2511720
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FD schemes for ODEs• Runge-Kutta is actually a family of methods!
• The most famous is Runge-Kutta-4 (RK4), which has p=4:
RK4 explicit
(ode45 in matlab) with
k1 =1
2f(Un, tn)
k2 =1
2f(Un +�tk1, tn+1/2)
k3 =1
2f(Un +�tk2, tn+1/2)
k4 =1
2f(Un + 2�tk3, tn+1)
Un+1 � Un
�t=
1
3[k1 + 2k2 + 2k3 + k4]
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Stability diagrams• So far, we obtained stability conditions such as (RK2):
• It makes sense (see lecture) to consider complex z, and draw the stability region in the complex plane
|1 + z +1
2z2| 1
6.2 Finite Difference Methods 469
Adams-Bashforth
5ii Adams-Moulton
Backward Differences Runae-Kutta
Figure 6.3: Stability regions IGI 5 1 for popular methods (stabi1ity.m on the cse site). Explicit methods are stable for a At inside the curves; implicit methods outside.
Runge-Kutta Methods
If evaluations of f (u, t ) are not too expensive, Runge-Kutta methods are highly competitive (and self-starting). Runge-Kutta of orders 4 and 5 is the basis for ode45. These are compound one-step methods, using Euler's Un + At fn inside f :
Simplified Runge- Kutta un+l-un 1 At = -[fn + f (Un + Atfn , tn+~)] . 2 (25)
You see the compounding of f . For u ' = au the growth factor G captures (At)2:
Comparing with the exact growth ea At, this confirms second-order accuracy. Stability hits a limit at aAt = -2 where G = 1. Now let a At = r be complex:
Stability limit for RK2 1 [GI = Il+x+-x21 = 1 for z = a + i b 2
The stability limit is a closed curve in the complex plane through x = a At = -2. Figure 6.3 shows all the numbers x (eigenvalues in the matrix case) at which IGI 5 1.
6.2 Finite Difference Methods 469
Adams-Bashforth
5ii Adams-Moulton
Backward Differences Runae-Kutta
Figure 6.3: Stability regions IGI 5 1 for popular methods (stabi1ity.m on the cse site). Explicit methods are stable for a At inside the curves; implicit methods outside.
Runge-Kutta Methods
If evaluations of f (u, t ) are not too expensive, Runge-Kutta methods are highly competitive (and self-starting). Runge-Kutta of orders 4 and 5 is the basis for ode45. These are compound one-step methods, using Euler's Un + At fn inside f :
Simplified Runge- Kutta un+l-un 1 At = -[fn + f (Un + Atfn , tn+~)] . 2 (25)
You see the compounding of f . For u ' = au the growth factor G captures (At)2:
Comparing with the exact growth ea At, this confirms second-order accuracy. Stability hits a limit at aAt = -2 where G = 1. Now let a At = r be complex:
Stability limit for RK2 1 [GI = Il+x+-x21 = 1 for z = a + i b 2
The stability limit is a closed curve in the complex plane through x = a At = -2. Figure 6.3 shows all the numbers x (eigenvalues in the matrix case) at which IGI 5 1.
(z = a Δt for u’=au)
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Basic ODE time integration• Given: RHS f(u,t), initial conditions u(0)
• Solve u’(t) = f(u,t):
Nsteps=1000; %Perform 1000 integration steps dt = 0.1;u = zeros(Nsteps+1); % Preallocate space for solution u % Set initial conditions: u(1)=u0; t = 0;
% Forward Euler: for i=1:1000
u(i+1) = u(i) + dt*f(u(i),t); t=t+dt;
end% Plot solution u etc.
If you use integrator that requires U from past: Start doing a single Forward Euler step to get the first 2 values of U, then use your integrator of choice!
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1-way wave equation
• So far we only considered ODE’s: u’ = f(u,t) and we only considered methods for the time integration
• Now: PDEs! Simplest example: 1-way wave equation:
• We need to use “good” time integration and good approximation of spatial derivatives.
ut
= cux
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1-way wave equation• Properties of :u
t
= cux
• Conservative equation (modes neither grow nor shrink)
lecture
• c is the speed by which the wave travels
• wave does not change its form
• Solution to initial condition : u(x, 0) = e
ikx
u(x, t) = e
ik(x+ct)
lecture
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Spatial finite difference (FD) schemes• Common FD schemes for 1D ux:
u(x)
x
Forward diff, O(Δx):
Backw. diff, O(Δx):i
Central diff, O(Δx2):
Lecture: accuracyBut this is only spatial accuracy! How does it affect overall accuracy?
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FD for the 1-way wave equation
Upwind/forward
differences
Lax-Wendroff
Lax-Friedrich
• Now: How to do time and space discretization in optimal way?
U(x, t+�t)� U(x, t)
�t
= c
U(x+�x, t)� U(x, t)
�x
U(x, t+�t)� U(x, t)
�t
= c
U(x+�x, t)� U(x��x, t)
2�x
+�t
2c
2U(x+�x, t)� 2U(x, t) + U(x��x, t)
�x
2
U(x, t+�t)� 12 (U(x+�x, t) + U(x��x, t))
�t
= c
U(x+�x, t)� U(x��x, t)
2�x
• Multiplying by △t gives various terms on RHS with factor r = c
�t
�x
CFL condition for stability: r≤1 (lecture)
until here
say next time why r<1 is stability criterium
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Courant-Friedrichs-Lewy condition
• The CFL condition is necessary but not sufficient for stability! r = c
�t
�x
1
• Sufficient criteria for stability: Neumann stability analysis
• Idea: Remember, wave equation has solution in terms of exponentials: => We found |G|=1 analytically, but how does this look for discrete approximations? => Will tell us stability for each mode k separately
u(x, t) = G(k, t)eikx
Neumann stab. analysis for upwing and Lax-Friedrichs: Lecture
0 r 1• We find: Upwind:Lax-Friedrichs: �1 r 1
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Lecture 418.086
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Last time
• 1-way wave equation (also called advection equation)
• CFL criterium for stability
• von Neumann stability analysis
• solution in terms of single modes:
r = c
�t
�x
1
ut
= cux
u(x, t) = G(k, t)eikx
• |G|=1 analytically• |G|<1 for stability of numerical scheme
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FD for 1-way wave eq: summary
Upwind differences
Lax-Wendroff
Lax-Friedrich
U(x, t+�t)� U(x, t)
�t
= c
U(x+�x, t)� U(x, t)
�x
U(x, t+�t)� U(x, t)
�t
= c
U(x+�x, t)� U(x��x, t)
2�x
+�t
2c
2U(x+�x, t)� 2U(x, t) + U(x��x, t)
�x
2
U(x, t+�t)� 12 (U(x+�x, t) + U(x��x, t))
�t
= c
U(x+�x, t)� U(x��x, t)
2�x
O(Δt)
O(Δt2)
O(Δt) !
0 r 1
�1 r 1
�1 r 1
r = c
�t
�x
Lecture
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Notation• We have seen different schemes for 1-way wave equation.
The schemes are often called stencils and depicted like this:
480 Chapter 6 Initial Value Problems
Lax-Friedrichs G matches the next term in the exact growth factor only if r2 = 1:
GLF = cos kAx + ir sin kAx = 1 + irkAx - AX)^ + . - - (22) 1 2 2 Gexact = e = 1 + irk Ax + 2i r (k AX)^ + - - -
In the exceptional cases r = 1 and r = -1, G agrees with Gexact. Staying exactly on the characteristic line, UJ,,+1 matches the true u ( j A x , t + At) . For r2 < 1, Lax- Friedrichs has an important advantage and disadvantage:
U,,,+1 is a positive combination of old values. But accuracy is only first-order.
4. Lax-Wendroff is stable for -1 < r 5 1. The LW difference equation combines U,,, and Uj-l,n and Uj+1,, to compute U,,n+l:
Lax- Wendroff 1 zk Ax 1 -zkAx G = (1 - r2) + -(r2 + r)e + -(r2 - r)e . 2 2 (23)
This is G = 1 - r2 + r2 cos kAx + ir sin kAx. At the dangerous frequency kAx = T ,
the growth factor is 1 - 2r2. That stays above -1 if r2 5 1. Problem 5 shows that IGI < 1 for every kAx. Lax-Wendroff is stable when-
ever the CFL condition r 2 < - 1 is satisfied. The wave can go either way (or both ways) since c can be negative. LW is the most accurate of these five methods.
upwind wrong way centered Lax-Friedrichs Lax-Wendroff stable unstable unstable stable stable i f r L 1 all At all At if Irl 5 1 if Irl 5 1
5. The centered methods of maximum accuracy are stable for -1 < r < 1. Lax-Wendroff uses three values at time level n for accuracy two. For every even p = 2q, there are p+ 1 coefficients a_,, . . . , a, so that the difference equation U,,n+l = C amU,+,,, has accuracy p. Matching G = C ameimkAx with the exact factor e - - eikrAx gives p + 1 equations C amm& rj for the a's.
Using all those values, the CFL condition allows the possibility of stability out to -q 5 c A t lAx 5 q. The actual requirement is -1 5 r 5 1.
Lax-Wendroff
Lax-Friedrich
U(x, t+�t)� U(x, t)
�t
= c
U(x+�x, t)� U(x��x, t)
2�x
+�t
2c
2U(x+�x, t)� 2U(x, t) + U(x��x, t)
�x
2
U(x, t+�t)� 12 (U(x+�x, t) + U(x��x, t))
�t
= c
U(x+�x, t)� U(x��x, t)
2�x
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Real-world accuracyUpwind:
R. J. LeVeque — AMath 585–6 Notes 183
0 1 2 3 4 5 6−0.5
0
0.5
1Upwind solution at time 4
0 1 2 3 4 5 6−0.5
0
0.5
1Lax−Wendroff solution at time 4
Figure 13.4: Numerical solution using upwind (diffusive) and Lax-Wendroff (dispersive) methods.
13.6.2 Lax-Wendroff
If the same procedure is followed for the Lax-Wendroff method, we find that all O(k) terms drop out ofthe modified equation, as is expected since this method is second order accurate on the advection equa-tion. The modified equation obtained by retaining the O(k2) term and then replacing time derivativesby spatial derivatives is
vt + avx +16ah2
!1 −
"ak
h
#2$
vxxx = 0. (13.37)
The Lax-Wendroff method produces a third order accurate solution to this equation. This equationhas a very different character from (13.35). The vxxx term leads to dispersive behavior rather thandiffusion. This is clearly seen in Figure 13.4, where the Un
j computed with Lax-Wendroff are comparedto the true solution of the advection equation. The magnitude of the error is smaller than with theupwind method for a given set of k and h, since it is a higher order method, but the dispersive termleads to an oscillating solution and also a shift in the location of the main peak, a phase error.
The group velocity for wave number ξ under Lax-Wendroff is
cg = a − 12ah2
!1 −
"ak
h
#2$ξ2
which is less than a for all wave numbers. (The concept of group velocity is explained in Section 13.7.)As a result the numerical result can be expected to develop a train of oscillations behind the peak, withthe high wave numbers lagging farthest behind the correct location.
If we retain one more term in the modified equation for Lax-Wendroff, we would find that the U nj
are fourth order accurate solutions to an equation of the form
vt + avx +16ah2
!1 −
"ak
h
#2$
vxxx = −ϵvxxxx, (13.38)
Lax-Wendroff:
R. J. LeVeque — AMath 585–6 Notes 183
0 1 2 3 4 5 6−0.5
0
0.5
1Upwind solution at time 4
0 1 2 3 4 5 6−0.5
0
0.5
1Lax−Wendroff solution at time 4
Figure 13.4: Numerical solution using upwind (diffusive) and Lax-Wendroff (dispersive) methods.
13.6.2 Lax-Wendroff
If the same procedure is followed for the Lax-Wendroff method, we find that all O(k) terms drop out ofthe modified equation, as is expected since this method is second order accurate on the advection equa-tion. The modified equation obtained by retaining the O(k2) term and then replacing time derivativesby spatial derivatives is
vt + avx +16ah2
!1 −
"ak
h
#2$
vxxx = 0. (13.37)
The Lax-Wendroff method produces a third order accurate solution to this equation. This equationhas a very different character from (13.35). The vxxx term leads to dispersive behavior rather thandiffusion. This is clearly seen in Figure 13.4, where the Un
j computed with Lax-Wendroff are comparedto the true solution of the advection equation. The magnitude of the error is smaller than with theupwind method for a given set of k and h, since it is a higher order method, but the dispersive termleads to an oscillating solution and also a shift in the location of the main peak, a phase error.
The group velocity for wave number ξ under Lax-Wendroff is
cg = a − 12ah2
!1 −
"ak
h
#2$ξ2
which is less than a for all wave numbers. (The concept of group velocity is explained in Section 13.7.)As a result the numerical result can be expected to develop a train of oscillations behind the peak, withthe high wave numbers lagging farthest behind the correct location.
If we retain one more term in the modified equation for Lax-Wendroff, we would find that the U nj
are fourth order accurate solutions to an equation of the form
vt + avx +16ah2
!1 −
"ak
h
#2$
vxxx = −ϵvxxxx, (13.38)
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Real-world accuracy478 Chapter 6 Initial Value Problems
1 2 -
1 .
08. Upwind 06.
04.
02.
0
Figure 6.6: Three approximations to a sharp signal show smearing and oscillation.
For an ideal difference equation, we only want dissipation very close to the shock. This can avoid oscillation without losing accuracy. A lot of thought has gone into high resolution met hods (Section 6.6), to capture shock waves cleanly.
Stability of the Four Finite Difference Methods
Accuracy requires G to stay close to the true eicbAt. Stability requires G to stay inside the unit circle. If IGI > 1 at frequency k, the solution Gnezkx will blow up.
We now check IGI < 1, in the four methods. CFL was only a necessary condition ! 1. Forward differences in space and time: AU/At = cAU/Ax (upwind). Equation (11) was G = 1 - r + reikAx. If the Courant number is 0 < r 5 1, then 1 - r and r will be positive. The triangle inequality gives IGI < 1:
Stability for 0 5 r < 1 zk Ax - IGl 5 11-rI+Ire I = 1 - r + r = l . (17)
This sufficient condition 0 5 c At/ Ax < 1 agrees with the CFL necessary condition U ( x , nAt) depends on the initial values between x and x + nAx. That domain of dependence must include the point x + c n At. (Otherwise, changing the initial value at the point x + c n A t would change the true solution u but not the approximation U.) Then 0 < c nAt < nAx means that 0 5 r 5 1.
Figure 6.7 shows G in the stable case r = and the unstable case r = % (when At is too large). As k varies, and eAAX goes around a unit circle, the complex number G = 1 - r + reikAx goes in a circle of radius r. The center is 1 - r. Always G = 1 at zero frequency (constant solution, no growth).
2. Forward difference in time, centered difference in space. This combination is never stable ! The shorthand U,,, will stand for U(jAx, nAt):
step-like initial condition:
Can we understand this behavior?
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Modified equations
• see lecture notes
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Real-world accuracyUpwind:
R. J. LeVeque — AMath 585–6 Notes 183
0 1 2 3 4 5 6−0.5
0
0.5
1Upwind solution at time 4
0 1 2 3 4 5 6−0.5
0
0.5
1Lax−Wendroff solution at time 4
Figure 13.4: Numerical solution using upwind (diffusive) and Lax-Wendroff (dispersive) methods.
13.6.2 Lax-Wendroff
If the same procedure is followed for the Lax-Wendroff method, we find that all O(k) terms drop out ofthe modified equation, as is expected since this method is second order accurate on the advection equa-tion. The modified equation obtained by retaining the O(k2) term and then replacing time derivativesby spatial derivatives is
vt + avx +16ah2
!1 −
"ak
h
#2$
vxxx = 0. (13.37)
The Lax-Wendroff method produces a third order accurate solution to this equation. This equationhas a very different character from (13.35). The vxxx term leads to dispersive behavior rather thandiffusion. This is clearly seen in Figure 13.4, where the Un
j computed with Lax-Wendroff are comparedto the true solution of the advection equation. The magnitude of the error is smaller than with theupwind method for a given set of k and h, since it is a higher order method, but the dispersive termleads to an oscillating solution and also a shift in the location of the main peak, a phase error.
The group velocity for wave number ξ under Lax-Wendroff is
cg = a − 12ah2
!1 −
"ak
h
#2$ξ2
which is less than a for all wave numbers. (The concept of group velocity is explained in Section 13.7.)As a result the numerical result can be expected to develop a train of oscillations behind the peak, withthe high wave numbers lagging farthest behind the correct location.
If we retain one more term in the modified equation for Lax-Wendroff, we would find that the U nj
are fourth order accurate solutions to an equation of the form
vt + avx +16ah2
!1 −
"ak
h
#2$
vxxx = −ϵvxxxx, (13.38)
Smeared out due to diffusive term in modified equations!
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Real-world accuracyLax-Wendroff:
R. J. LeVeque — AMath 585–6 Notes 183
0 1 2 3 4 5 6−0.5
0
0.5
1Upwind solution at time 4
0 1 2 3 4 5 6−0.5
0
0.5
1Lax−Wendroff solution at time 4
Figure 13.4: Numerical solution using upwind (diffusive) and Lax-Wendroff (dispersive) methods.
13.6.2 Lax-Wendroff
If the same procedure is followed for the Lax-Wendroff method, we find that all O(k) terms drop out ofthe modified equation, as is expected since this method is second order accurate on the advection equa-tion. The modified equation obtained by retaining the O(k2) term and then replacing time derivativesby spatial derivatives is
vt + avx +16ah2
!1 −
"ak
h
#2$
vxxx = 0. (13.37)
The Lax-Wendroff method produces a third order accurate solution to this equation. This equationhas a very different character from (13.35). The vxxx term leads to dispersive behavior rather thandiffusion. This is clearly seen in Figure 13.4, where the Un
j computed with Lax-Wendroff are comparedto the true solution of the advection equation. The magnitude of the error is smaller than with theupwind method for a given set of k and h, since it is a higher order method, but the dispersive termleads to an oscillating solution and also a shift in the location of the main peak, a phase error.
The group velocity for wave number ξ under Lax-Wendroff is
cg = a − 12ah2
!1 −
"ak
h
#2$ξ2
which is less than a for all wave numbers. (The concept of group velocity is explained in Section 13.7.)As a result the numerical result can be expected to develop a train of oscillations behind the peak, withthe high wave numbers lagging farthest behind the correct location.
If we retain one more term in the modified equation for Lax-Wendroff, we would find that the U nj
are fourth order accurate solutions to an equation of the form
vt + avx +16ah2
!1 −
"ak
h
#2$
vxxx = −ϵvxxxx, (13.38)
Dispersion due to term vxxx in modified equations
High freq. modes travel slower (=> on the left of the envelope)
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Phase vs. group velocity• Remember from physics:
R. J. LeVeque — AMath 585–6 Notes 187
−3 −2 −1 0 1 2 3−1
−0.5
0
0.5
1
time = 0
−3 −2 −1 0 1 2 3−1
−0.5
0
0.5
1
time = 0.4
−3 −2 −1 0 1 2 3−1
−0.5
0
0.5
1
time = 0.8
−3 −2 −1 0 1 2 3−1
−0.5
0
0.5
1
time = 1.2
−3 −2 −1 0 1 2 3−1
−0.5
0
0.5
1
time = 1.6
Figure 13.6: The oscillatory wave packet satisfies the dispersive equation ut + aux + buxxx = 0. Alsoshown is a black dot, translating at the phase velocity cp(ξ0) and a Gaussian that is translating at thegroup velocity cg(ξ0).
phase velocity
group velocity
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Dispersion in LW schemeR. J. LeVeque — AMath 585–6 Notes 187
−3 −2 −1 0 1 2 3−1
−0.5
0
0.5
1
time = 0
−3 −2 −1 0 1 2 3−1
−0.5
0
0.5
1
time = 0.4
−3 −2 −1 0 1 2 3−1
−0.5
0
0.5
1
time = 0.8
−3 −2 −1 0 1 2 3−1
−0.5
0
0.5
1
time = 1.2
−3 −2 −1 0 1 2 3−1
−0.5
0
0.5
1
time = 1.6
Figure 13.6: The oscillatory wave packet satisfies the dispersive equation ut + aux + buxxx = 0. Alsoshown is a black dot, translating at the phase velocity cp(ξ0) and a Gaussian that is translating at thegroup velocity cg(ξ0).
R. J. LeVeque — AMath 585–6 Notes 187
−3 −2 −1 0 1 2 3−1
−0.5
0
0.5
1
time = 0
−3 −2 −1 0 1 2 3−1
−0.5
0
0.5
1
time = 0.4
−3 −2 −1 0 1 2 3−1
−0.5
0
0.5
1
time = 0.8
−3 −2 −1 0 1 2 3−1
−0.5
0
0.5
1
time = 1.2
−3 −2 −1 0 1 2 3−1
−0.5
0
0.5
1
time = 1.6
Figure 13.6: The oscillatory wave packet satisfies the dispersive equation ut + aux + buxxx = 0. Alsoshown is a black dot, translating at the phase velocity cp(ξ0) and a Gaussian that is translating at thegroup velocity cg(ξ0).