Math C42H 3042: Exercise Sheet 1

1. The shallow water equations can be reduced to
   where d=d(x,t) is the water depth and v=v(x,t) is the horizontal (x) fluid velocity. Derive a linear system by assuming that d(x,t)=d₀ + eps d₁(x,t), v(x,t)=v₀ + eps v₁(x,t) with eps << 1 and d₀ and v₀ are constant and neglecting terms less than O(eps). Show the system is strictly hyperbolic. Find the exact solution that evolves from the initial data d₁⁰(x) = 0 for |x| > X,
   d₁⁰(x) = 1 for |x| < X
   v₁⁰(x) = 0 for all x

2. The exact solution of u_t + a u_x = 0 satisfies u(x_j,t_n+1)=u(x_j-ak,t_n) where t_n+1=t_n+k. Show that the first-order upwinding, Lax-Friedrichs, Lax-Wendroff and Beam-Warming schemes can be derived by fitting suitable interpolation polynomials to a subset of {Uⁿ} about x_j and tracing characteristics back from (x_j,t_n+1).

3. Use a direct 1-norm stability argument to find a sufficient stability condition for the scalar Lax-Friedrichs scheme. Compare this with the CFL condition for the scheme. Comment on the stability of Lax-Friedrichs schemes for linear systems.

4. Determine the LTE of the scalar upwind, Lax-Wendroff and Beam-Warming schemes. Comment on the necessary smoothness of the exact solution in each case.

5. Perform convergence and relative error numerical experiments on the first order upwinding, Lax-Friedrichs, Lax-Wendroff and Beam-Warming schemes for the scalar advection equation. Use the example matlab codes as the basis for these experiments. Compare the 1-, 2- and infinity-norm of the numerical solution error at a given fixed time, T for various values h with Courant number, v, constant and for various v with h constant. Repeat the experiments with discontinuous data.

6. Use von Neumann stability analysis to find the stability restrictions of Lax-Friedrichs and Beam-Warming schemes.

7. Use von Neumann stability analysis to find a stability restriction for the explicit scheme
   Uⁿ⁺¹_j = Uⁿ_j + kD( Uⁿ_j+1 - 2 Uⁿ_j + Uⁿ_j-1 )/h²
   for the heat equation u_t = D u_xx.

8. Find the numerical solution to Q.1 using both Lax-Wendroff and symmetrized (first order) upwinding and Beam-Warming schemes. Take d₀=1, v₀=0.5 and X=0.05.

9. What happens if you choose v = ak/h = 1 for scalar upwinding, Lax-Friedrichs or Lax-Wendroff schemes? (Note that this ``exact'' behaviour is unstable due to round-off errors and does not apply to systems or non-linear equations.)