HW 4.2.1 Write a code to solve the heat equation,
(4.2.10)-(4.2.13) with the explicit scheme (4.2.5)-(4.2.9) where nu = 1.
Use Mx = My = 20, Delta t = 0.0005 and compare true and approximate
solutions at t = 0.06, 0.1, and 0.9. Next use Delta t = 0.001 and
finally use Mx = 10 and My = 20.
Problem #2
HW 4.2.7 Write a code to solve the heat equation,
(4.2.14)-(4.2.17) with the explicit scheme (4.2.5)-(4.2.9) where nu = 1.
Note the Neumann boundary condition. Use Mx = My = 10, Delta t = 0.001
and compare true and approximate solutions at t = 0.06, 0.1, and 0.9.
Use a second-order approximation to the Neumann boundary condition.
Problem #3
Show that the explicit scheme (4.2.5) is consistent of order (2,2,1)
(thus O(Delta x^2) + O(Delta y^2) + O(Delta t)) both pointwise and in
sup-norm.
Problem #4
HW 4.3.2 Discuss the stability of the scheme in HW 4.3.2 for the
two-dimensional pure initial-value advection-diffusion equation.
Problem #5
HW 4.3.4 Discuss the stability of the two-dimensional
Crank-Nicolson scheme for the initial-boundary-value heat equation.
Problem #6
HW 4.4.11 Show that the three-dimensional Peaceman-Rachford scheme in
(4.4.102)-(4.4.104) is conditionally stable.
Problem #7
HW 5.3.3 Verify that the explicit MacCormack scheme is equivalent
to the explicit Lax-Wendroff scheme (hence stability and consistency
results follow).
Problem #8
HW 5.4.5 Verify the consistency and stability properties of the
implicit Lax-Wendroff scheme.
Problem #9
HW 5.7.1 Use the CFL condition to obtain necessary conditions for
the convergence of the Lax-Friedrichs, Beam-Warming, and MacCormack
methods.
Problem #10
HW 5.8.1 For the upwind scheme applied to the two-dimensional
one-way wave equation plot the modulus of the symbol for R_x and R_y
satisfying R_x + R_y < 1. Assume that a and b are greater than zero.
Problem #11
HW 6.2.2 Analyze the properties of the standard implicit method
for the hyperbolic system of partial differential equations given in
(6.2.38).
Problem #12
HW 6.2.6 Analyze the stability of the explicit flux split scheme
(6.2.61).
Problem #13
HW 6.3.1 Determine the assignment of the boundary conditions for
the hyperbolic system (6.3.32).
Problem #14
HW 6.3.5 Use the FTFS scheme to approximate the solution to the
hyperbolic system (6.3.32) in Problem #13 using the initial conditions
v_1(x,0) = 2e^{x-1}, v_2(x,0) = sin(pi x), and M = 100.