Write a code to solve the heat equation, (1.1)-(1.3) with the explicit
scheme (1.10)-(1.13) where f(x) = sin(2 pi x), a(t) = b(t) = 0, M = 10,
and nu = 1/6. Use Delta t = 0.02 and graph true and approximate
solutions at t = 0.06, 0.1, and 0.9. Experiment with the values of M
(and hence Delta x) and Delta t. Then run your code to t = 50.
Problem #2
Verify the second order approximation to the first derivative given in
(1.19).
Problem #3
Verify the second order approximation to the second derivative given in
(1.20).
Problem #4
Write a code to solve the heat equation, (1.1)-(1.3) with the leapfrog
scheme (1.25), (1.11)-(1.13) where f(x) = sin(2 pi x), a(t) = b(t) = 0, M = 10,
and nu = 1/6. Use Delta t = 0.02 and graph true and approximate
solutions at t = 0.06, 0.1, and 0.9. Experiment with the values of M
(and hence Delta x) and Delta t. Use the exact solution to generate
additional starting values along the time level t_1 = Delta t.
Problem #5
Use the ideal choice of Delta t = Delta x^2/(6 nu) in your explicit
scheme code of Problem #1. Experiment with different choices of M
(changing Delta x) which will change Delta t accordingly. Discuss your
results.
Problem #6
Write a code to solve the heat equation, (1.32)-(1.35) (Neumann boundary
conditions) with the explicit scheme (1.10), (1.11), (1.13), and both
the first order (1.39) and the second order (1.41) approximations to the
value of u_0^{n+1}. Let f(x) = cos(pi x/2), a(t) = b(t) = 0, M = 10,
and nu = 1. Use Delta t = 0.004 and graph true and approximate
solutions at t = 0.06, 0.1, and 0.9. Experiment with the values of M
(and hence Delta x) and Delta t.
Problem #7
Write a code to solve the advection-diffusion equation, (1.42)-(1.44)
with the explicit scheme (1.45)-(1.48) where f(x) = sin(4 pi x), a(t) =
b(t) = 0, M = 20, nu = 1, and a = 2. Use Delta t = 0.001 and graph true
and approximate solutions at t = 0.06, 0.1, and 0.9. Experiment with
the values of M (and hence Delta x) and Delta t. Then run your code
with nu = 0.01.
Problem #8
Write a code to solve the nonhomogeneous heat equation, (1.49)-(1.52)
with the explicit scheme (1.53), (1.46)-(1.48). Let f(x) = x(1-x),
a(t) = 10 sin(t), b(t) = 4 sin(6t), F(x,t) = sin(2 pi x) sin(4 pi t), M
= 10, and nu = 0.1. Use Delta t = 0.05 and graph true and approximate
solutions at t = 0.1, 0.9, and 2.0. Experiment with the values of M
(and hence Delta x) and Delta t.
Problem #9
Use the method of undetermined coefficients to develop a second order
approximation to the fourth partial derivative, v_{xxxx}.
Problem #10
Derive the leapfrog scheme using the conservation law approach and
midpoint quadratures.
Problem #11
Show the scheme in (1.80) is consistent to order O(Delta t) +
O(Delta x) to the partial differential equation v_t = (9 nu/8) v_{xx}.
Problem #12
Find O((Delta x)^2) approximations to (v_x)_{k+1/2} and (v_x)_{k-1/2}
using v_{k-1}, v_k, and v_{k+1}. What scheme results from the
conservation law approach with these approximations?
Problem #13
Prove that the solution to the Lax-Friedrichs scheme converges in
sup-norm to the solution of the wave equation for |R| < or = 1 where R =
a Delta t / Delta x.
Problem #14
Use your code to solve the heat equation, (1.1)-(1.3) with the explicit
scheme (1.10)-(1.13) where f(x) = sin(4 pi x), a(t) = b(t) = 0, and nu =
0.1. In particular, look at the solution at t = 0.05 and t = 0.1. (a)
Use Delta x = 0.1, Delta t = 0.05; (b) Delta x = 0.05, Delta t = 0.0125;
(c) Delta x = 0.01, Delta t = 0.0005. Compare and contrast your results.
Problem #15
Determine the order of accuracy of the explicit scheme for
the pure initial-value advection-diffusion equation.
Problem #16
Determine the order of accuracy of the Crank-Nicolson scheme for the
pure initial-value heat equation.
Problem #17
Use your code to solve the heat equation, (1.32)-(1.35) (Neumann boundary
conditions) with the explicit scheme, either (i) (2.3.19)-(2.3.21), (ii)
(2.3.30)-(2.3.32), or (iii) (2.3.30)-(2.3.32) with an offset grid. Let
f(x) = cos(pi x/2), a(t) = b(t) = 0, and nu = 1. (a) Use M = 20, Delta
t = 0.001; (b) M = 40, Delta t = 0.00025. Compare and contrast your
results.
Problem #18
Determine the order of accuracy of the explicit scheme for the
initial-boundary-value heat equation given in HW 2.3.5 (a) in the book
with a = 0.
Problem #19
Show that for |R| less than or equal to 1 (where a < 0), the scheme
(2.4.23)-(2.4.25) for the one way IBVP wave equation (2.4.20)-(2.4.22)
is stable.
Problem #20
Analyze the stability and convergence of the downwind (FTBS) scheme for
the pure initial-value problem, one way wave equation with negative wave
speed. Use the tools of discrete Fourier transforms.