Page 189 - Computational Fluid Dynamics for Engineers

176 5. Numerical Methods for Model Hyperbolic Equations

5-11. Solve Eq. (5.1.1) subject to the initial and boundary conditions

( 1 x < 10
u(x,Q) = \ (P5.11.1)
[ 0 x > 10
and
u(0,t) = 1 (P5.11.2)
by using the Lax-Wendroff method for At/Ax = 1.0, 0.6 and 0.3. Take c = 1,
choose a 41 grid point mesh with Ax = 1 and compute up to t — 18. Compare
your solutions with the exact solution (P5.10.1) graphically.

5-12. Repeat Problem 5.11 with the MacCormack method.

5-13. Repeat Problem 5.11 with the Beam-Warming method.

5-14. Repeat Problem 5.11 with the upwind method Eq. (5.5.3).

5-15. Repeat Problem 5.11 with the upwind method Eq. (5.5.26).

5-16. Show that
2
u(x, t) = exp(-k vt) sin[k(x - ct)} (P5.16.1)
is a solution of the linearized Burger's equation.

ut + cu x = vu xx (P5.16.2)
for the initial condition

u(x, 0) = sin(fcx) 0 < x < 2n (P5.16.3)

and the periodic boundary condition.
u(0,t) =u{2n,t) (P5.16.4)

5-17. Use the Lax-WendrofF method to solve Eq. (P5.16.2-4) with k = 2, c = \
4
3
2
for values of v = 10~\ 10~ , 10~ and 10~ . Take At/Ax = 1.0 and At = 0.02.
Compare your solution graphically with the exact solution.
5-18. Repeat Problem 5.17 with the MacCormack method.

5-19. Repeat Problem 5.17 with the Beam-Warming method.

5-20. Repeat Problem 5.17 with the upwind method Eq. (5.5.3).

5-21. Repeat Problem 5.17 with the upwind method Eq. (5.5.26).