Write a code to solve the heat equation with (a) the explicit scheme and
(b) the implicit scheme, where f(x) = sin(pi x), a(t) = b(t) = 0, and
nu = 1/6. Graph true and approximate solutions at t = 0.001, 0.06, 0.1,
0.9, and 50.0. Experiment with the values of M (and hence Delta x) and
Delta t.
Problem #2
Using the definitions, analyze convergence, consistency, and stability
for both the FTFS and FTCS schemes for the one-way wave equation
initial-value problem.
Problem #3
Using the discrete Fourier transform, analyze stability for the FTFS,
BTFS, and BTCS schemes for the one-way wave equation initial-value
problem.
Problem #4
Analyze the stability for the FTFS scheme applied to the one-way wave
equation initial-boundary-value problem (with negative wave speed a).
Problem #5
Write codes to solve the one-way wave equation on 0 < x < 1 with
periodic boundary conditions with (a) the FTBS scheme, (b) the
Lax-Friedrichs scheme, and (c) the Lax-Wendroff explicit
scheme. The initial data is f(x) = f1(x) or f2(x) where f1(x) is 1 on
[0.4,0.6] and zero elsewhere while f2(x) is 1 on [0,0.2] and zero
elsewhere. Let a = 1. Graph true and approximate solutions at t =
0.4, 0.5, 0.6, 1.0, and 2.0. In particular, try M = 100 and R = 0.8.
Problem #6
Analyze the dissipation and dispersion for the Lax-Wendroff scheme
applied to the one-way wave equation initial-boundary-value problem
(with positive wave speed a). Use both asymptotic and graphical
techniques.
Problem #7
Use your Lax-Wendroff code to solve the one-way wave equation on
0 < x < 1 with periodic boundary conditions. The initial data is f(x) =
sin^{40}(pi*x). Let a = 1. Use all the techniques we have developed to
investigate the features of this scheme.
Problem #8
Analyze the dissipation and dispersion for the Lax-Wendroff scheme
applied to the one-way wave equation initial-boundary-value problem.
Use the modified partial differential equation.
Problem #9
Write codes to solve the linear advection-diffusion equation on 0 < x < 1
with homogeneous Dirichlet boundary conditions with (a) the centered
explicit scheme, (b) the Lax-Wendroff explicit scheme, and (c) the
Crank-Nicolson scheme. The initial data is f(x) = f1(x) or f2(x) where
f1(x) is 1 on [0.4,0.6] and zero elsewhere while f2(x) = sin(2*pi*x).
Let a = -1 and try both nu = 1 and nu = 0.00001. Graph approximate
solutions at t = 0.4, 0.5, 0.6, 1.0, and 2.0. In particular, try M = 100.
Problem #10
Write codes to solve the nonlinear Burgers' equation on 0 < x < 1
with homogeneous Dirichlet boundary conditions with (a) the centered
explicit scheme, (b) the Lax-Wendroff explicit scheme, and (c) the
Crank-Nicolson scheme. The initial data is f(x) = f1(x) or f2(x) where
f1(x) is 1 on [0.4,0.6] and zero elsewhere while f2(x) = sin(2*pi*x).
Try nu = 1, nu = 0.00001, and nu = 0. Graph approximate
solutions at t = 0.1, 0.2, 0.3, 0.4, and 1.0. In particular, try M = 100.