your codes and descriptions on the appropriate runs you made by following Grader’s instructions.
Write your own code, individually. Do not copy codes!
- Consider the Heat Equation in the unit square [0,1] × [0,1] with initial condition
(
ut = D(uxx + uyy)
u(0, x, y) = sin(πx) sin(πy)
(1)
and the homogeneous Dirichlet boundary condition u = 0.
(a) Implement the Peaceman-Rachford ADI method to find an approximation to this initialboundary value problem.
(b) Accuracy check. To verify the accuracy of a scheme and as a check for possible
bugs one should always do a resolution study. Suppose that v(k) is your numerical
approximation at fixed time computed using time-step size k, then second order in time
accuracy means that
v(k) = u + c2k
2 + c3k
3 + · · · , (2)
where u is the exact value. Keeping the spatial resultion fixed (hx = hy = h, uniform
grid) now compute using k/2 then your approximation satisfies:
v(k/2) = u +
1
4
c2k
2 +
1
8
c3k
3 + · · · , (3)
then for v(k/4), etc. Then the ratios:
R(k) = v(k) − v(k/2)
v(k/2) − v(k/4) (4)
(c) Compute solution for D = 1 and plot it for at three different times.
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1 - Consider the one-way wave equation ut + ux = 0 on the interval [−1, 3] and for t ≥ 0 with
the following two sets of initial conditions:
u(x, 0) = (
1 − |x| if |x| ≤ 1,
0 otherwise,
(5)
and
u(x, 0) = e
−5x
2
. (6)
(a) Use the forward-time forward-space scheme:
u
n+1
j − u
n
j
k
+
u
n
j+1 − u
n
j
h
= 0,
with right-point boundary condition u
n+1
M = u
n+1
M−1 where xM = 3 to compute an approximation to the solution at several (up to 40) time steps. Use h = 0.02 and λ = k/h = 0.8.
Demonstrate numerically (plot the solution) the instability of the scheme and show that
the instability appears sooner with the less smooth initial data.
(b) Comment on the localization of the onset of instability for initial data (5) and give an
estimate of the expected growth rate of the instability per time step.
(c) Using the left boundary condition u(−1) = 0, write a stable scheme and compute the
corresponding approximation for data sets (5) and (6). Plot the approximations at
representative time steps. Use again h = 0.02 and λ = k/h = 0.8.
Sample Solution
