Page 187 - Computational Fluid Dynamics for Engineers

174 5. Numerical Methods for Model Hyperbolic Equations

[5] Jameson, A., Schmidt, W. and Turkel, E.: Numerical simulation of the Euler equations
by finite volume methods using Runge-Kutta time-stepping schemes, AIAA Paper 81-
1259, 1981.
[6] Steger, J.L. and Warming, R. F.: Flux vector splitting of the inviscid gas dynamic
equations with applications to finite-difference methods. J. Comput. Phys. 40, 263-
293, 1981.
[7] Osher, S. and Solomon, F.: Upwind difference schemes for hyperbolic systems of con-
servation laws. Math. Comput 38(158), 339-374, 1982.
[8] Roe, P. L.: Approximate Riemann solvers, parameter vectors, and difference schemes.
J. Comput Phys. 43, 357-372, 1981.

Problems

5-1. Solve Eq. (5.1.1) subject to the following boundary and initial conditions
x = 0, u = 0
f sin 27rx 0 < x < 1
t = 0, u= < _
\0 1 < x < 5
by the two-step Lax-Wendroff method for Ax = 0.1 at t = 4 for At = 0.01, 0.1
and 0.2. Take c= 1. Compare your solutions with the analytical solution

f sin27r( x-t) t<x<t+l
otherwise

and determine the percentage error. To implement the numerical boundary
condition, use the first-order backward difference formula (4.3.5), u$ — u\ = 0.

5-2. Repeat Problem 5.1 using the Crank-Nicolson method.

5-3. Solve the inviscid Burger's equation, Eq. (4.2.8), subject to the boundary
and initial conditions
u(0,t) = 0, t>0
u(x,0) = x, 0 < x < 1
by the McCormack method. Take Ax = 0.02 and At = 0.01 and perform the
calculations for 200 time steps.

5-4. Repeat Problem 5.3 using the Beam-Warming method (trapezoidal scheme
6 — ?£, £ = 0) and the Euler implicit scheme. Experiment with different CFL
numbers.

5-5. Show that for one-dimensional flow, the Jacobian matrix A of the flux
vector E can be written in the form given by Eq. (5.1.5c).

Hint: Eliminate p in Eq. (5.1.3b) with the expression given by Eq. (P2.16.1) for
one-dimensional flow.