Math 6630, Spring 2016, Problem Set 1
Due – Friday Feburary 12, 2016.
1) We often use the eigenvalues and eigenvectors of the discrete Laplacian operator and
being able to derive them is important. One way to do this uses the theory of solutions of
linear difference equations. Recall that an equation
k
X
aj yn+j = 0
(1)
j=0
is called a kth order linear constant coefficient difference equation, and that it has solutions
yn = rn where r satisfies
k
X
aj rj = 0.
(2)
j=0
If the k roots r1 , r2 , ..., rk of (2) are distinct, the general solution of (1) has the form
yn =
k
X
cj rjn
(3)
j=1
for constants c1 , c2 , ..., ck . Consider the discretization
−h−2 (uj−1 − 2uj + uj+1 ) = fj ,
j = 1, ..., N
u0 = uN +1 = 0,
(4)
with (N + 1)h = 1 for −v 00 (x) = f (x) on 0 ≤ x ≤ 1, with boundary conditions v(0) = v(1) =
0. The corresponding eigenvalue problem/eigenvector problem is to find x = (x1 , ..., xN )T 6=
0 and λ such that
−h−2 (xj−1 − 2xj + xj+1 ) = λxj ,
j = 1, ..., N
x0 = xN +1 = 0.
(5)
a) Use the theory described above for solutions of linear difference equations to determine
the eigenvalues and eigenvectors for this system.
b) Do the same thing for the same difference scheme but with discrete Neumann boundary
conditions (u1 − u0 )/h = 0 and (uN +1 − uN )/h = 0.
2) Show that the upwind scheme and Lax-Wendroff scheme for vt + cvx = 0 (with c > 0) can
be derived as follows. To determine ujn+1 , trace the characteristic curve in the xt-plane for
the PDE back to time level tn . If the CFL condition holds (assume it does), the intersection
of the characteristic with the line t = tn will occur between xj−1 and xj . The solution of
the PDE at (xj , tn+1 ) is the value of the solution of the PDE at this intersection point.
But we don’t know the value at this point if it is not a grid point. Show that the upwind
scheme is obtained if this value is interpolated from (xj−1 , unj−1 ) and (xj , unj ), and that the
1
Lax-Wendroff scheme is obtained if the value is interpolated from (xj−1 , unj−1 ), (xj , unj ), and
(xj+1 , unj+1 ).
3)
vt = vxx , 0 < x < 1
v(0, t) = 1, v(1, t) = 0
(
1 if x < 0.5
v(x, 0) =
0 if x ≥ 0.5
1. Use Crank-Nicolson with grid spacing h = 0.02 and time step 0.1 to solve the problem
up to time t = 1. Comment on your results. What is wrong with this solution?
2. Give a mathematical argument to explain the unphysical behavior you observed in the
numerical solution.
3. Experiment with smaller time steps. How small does the time step need to be to get
reasonable results?
4. What happens to the numerical solution as k → 0 with the ratio k/h fixed? Explain.
Would this same behavior occur using backward Euler in place of Crank-Nicolson?
Explain.
4) Derive the modified equation for the Lax-Friedrich’s scheme for vt + cvx = 0 through
terms of first order. Compare this modified equation to that for the upwind scheme and
make predictions about their relative performance.
5) Write programs to solve the advection equation
vt + cvx = 0,
on [0, 1] with periodic boundary conditions using upwinding and Lax-Wendroff. For smooth
solutions we expect upwinding to be first-order accurate and Lax-Wendroff to be secondorder accurate, but it is not clear what accuracy to expect for nonsmooth solutions.
1. Let a = 1 and solve the problem up to time t = 1. Perform a refinement study for both
upwinding and Lax-Wendroff with k = 0.8h with a smooth initial condition. Compute
the rate of convergence in the 1-norm, 2-norm, and max-norm. Note that the exact
solution at time t = 1 is the initial condition, and so computing the error is easy.
2. Repeat the previous problem with the discontinuous initial condition
(
1 if |x − 1/2| < 1/4
v(x, 0) =
0 otherwise
2
NOTE: See Appendix A LeVeque’s book for information on carrying out a convergence
study as asked for in this problem.
6) Consider three-point explicit schemes for the linear advection equation in the real line of
the form
un+1
= unj − C unj − unj−1 + D unj+1 − unj .
j
Show that
X
X n
un+1 − un+1 ≤
uj − unj−1 j
j−1
j
(6)
j
if C ≥ 0, D ≥ 0, and C + D ≤ 1. When the numerical solution of a scheme satisfies (6)
the scheme is total variation diminishing or TVD. Put upwinding and Lax-Wendroff into
the above form, and show that upwinding is TVD when it is stable and that Lax-Wendroff
is not TVD. Give an interpretation for the meaning of TVD and explain how this relates to
the numerical solutions from problem 5.
7) Consider the forward time, centered space discretization
un+1
− unj
unj+1 − unj−1
unj−1 − 2unj + unj+1
j
+a
=b
,
k
2h
h2
to the convection-diffusion equation,
vt + avx = bvxx , b > 0.
1. Let ν = ak/h and µ = bk/h2 . Use Fourier analysis to show that the scheme is stable
if µ ≤ 1/2.
2. Let a = 80, b = 1, h = 0.05. Generate a numerical solution on the spatial domain
[0, 1] with periodic boundary conditions using k = 0.25h2 /b with initial condition
v(x, 0) = exp(−20(x − 0.5)2 ). What happens? Does your stability analysis predict
this?
3. Since the solution to the PDE does not grow in time, it seems reasonable to require
that the numerical solution not grow in time. Show that the numerical solution does
not grow (in 2-norm) if and only if ν 2 ≤ 2µ ≤ 1. This is called strict or practical
stability, and as the name suggests it is the restriction one would use in practice.
4. Generate a numerical solution up to time t = 10−2 .
3