Problems
301
(a) Derive an expression for 8 2 1/1/8t
2 in tenns of the spatial derivatives of
1/1 and functions of the velocity field.
(b) Show that a fully second-order Lax-Wendroff approximation to this
problem can be obtained using (5.41) with c replaced by (c n + 1 + c n )/ 2.
12. Show that the antidiffusion step (5.60) of the Smolarkiewicz positive definite advection scheme is not monotone.
13. Suppose that f (s) is a continuously differentiable function of s and that
1/I(x, t) is a solution to the scalar conservation law (5.6). Show that the characteristic curves for this hyperbolic partial differential equation are straight
lines.
14. *Compute solutions to the advection equation (5.18) on the periodic domain 0 ::: x ::: I subject to the initial condition 1/1(x, 0) = sin" (2n x). Let
c = 0.1.
(a) Compare the exact solution with numerical solutions obtained using forward, Lax-Wendroff, and flux-limited methods. In the flux-limited methods
compute the low-order flux using the upstream scheme and the high-order
flux using the Lax-Wendroff method, but try three different flux limiters:
the MC, the minmod, and the superbee. Perform the simulations using a
Courant number cSt /!1x = 0.5 andAr = 1/40. As part of your discussion submit two plots of the solution at time t = 20, one comparing the
exact solution with that obtained using the three different flux limiters, and
one comparing the exact , upstream, Lax-Wendroff, and MC flux-limited
solutions. Scale the vertical axis so that -0.4 ::: 1/1 (x) ::: 1.4.
(b) Repeat the preceding simulations for the initial condition
1/I(x,O) = {I iflx -.11::: i;
o otherwise.
Again submit two plots of the solution at time t = 20, one comparing the
exact solution with that obtained using the three different flux limiters, and
one comparing the exact, upstream, Lax-Wendroff, and MC flux-limited
solutions. Discuss your results.
15. *Detennine the effective order of accuracy of the minmod, MC, and superbee flux-limited approximations to the advection equation considered in
Problem 14 except use the very smooth initial data 1/1 (x, 0) = sin(2nx).
In addition, compute results for the Zalesak FCT method using upstream
differencing for the low-order solution and the Lax-Wendroff scheme for
the higher-erder solution. Also try the iterative FCT scheme discussed at
the end of Section 5.4.2 using the preceding noniterated FCT solution for
the low-order scheme during the second iteration. Keep the Courant number
fixed at 0.5, and use !1x = 1/20, 1/40, 1/80, 1/160, and 1/320. Compute the
301
(a) Derive an expression for 8 2 1/1/8t
2 in tenns of the spatial derivatives of
1/1 and functions of the velocity field.
(b) Show that a fully second-order Lax-Wendroff approximation to this
problem can be obtained using (5.41) with c replaced by (c n + 1 + c n )/ 2.
12. Show that the antidiffusion step (5.60) of the Smolarkiewicz positive definite advection scheme is not monotone.
13. Suppose that f (s) is a continuously differentiable function of s and that
1/I(x, t) is a solution to the scalar conservation law (5.6). Show that the characteristic curves for this hyperbolic partial differential equation are straight
lines.
14. *Compute solutions to the advection equation (5.18) on the periodic domain 0 ::: x ::: I subject to the initial condition 1/1(x, 0) = sin" (2n x). Let
c = 0.1.
(a) Compare the exact solution with numerical solutions obtained using forward, Lax-Wendroff, and flux-limited methods. In the flux-limited methods
compute the low-order flux using the upstream scheme and the high-order
flux using the Lax-Wendroff method, but try three different flux limiters:
the MC, the minmod, and the superbee. Perform the simulations using a
Courant number cSt /!1x = 0.5 andAr = 1/40. As part of your discussion submit two plots of the solution at time t = 20, one comparing the
exact solution with that obtained using the three different flux limiters, and
one comparing the exact , upstream, Lax-Wendroff, and MC flux-limited
solutions. Scale the vertical axis so that -0.4 ::: 1/1 (x) ::: 1.4.
(b) Repeat the preceding simulations for the initial condition
1/I(x,O) = {I iflx -.11::: i;
o otherwise.
Again submit two plots of the solution at time t = 20, one comparing the
exact solution with that obtained using the three different flux limiters, and
one comparing the exact, upstream, Lax-Wendroff, and MC flux-limited
solutions. Discuss your results.
15. *Detennine the effective order of accuracy of the minmod, MC, and superbee flux-limited approximations to the advection equation considered in
Problem 14 except use the very smooth initial data 1/1 (x, 0) = sin(2nx).
In addition, compute results for the Zalesak FCT method using upstream
differencing for the low-order solution and the Lax-Wendroff scheme for
the higher-erder solution. Also try the iterative FCT scheme discussed at
the end of Section 5.4.2 using the preceding noniterated FCT solution for
the low-order scheme during the second iteration. Keep the Courant number
fixed at 0.5, and use !1x = 1/20, 1/40, 1/80, 1/160, and 1/320. Compute the