Math C42H 3042: Exercise Sheet 1

- The shallow water equations can be reduced to

whered=d(x,t)is the water depth andis the horizontal (v=v(x,t)x) fluid velocity. Derive a linear system by assuming thatd(x,t)=d,_{0}+ eps d_{1}(x,t)v(x,t)=vwith_{0}+ eps v_{1}(x,t)eps << 1anddand_{0}vare constant and neglecting terms less than_{0}O(eps). Show the system is strictly hyperbolic. Find the exact solution that evolves from the initial datadfor_{1}^{0}(x) = 0|x| > X,

dfor_{1}^{0}(x) = 1|x| < X

vfor all_{1}^{0}(x) = 0x

- The exact solution of
usatisfies_{t}+ a u_{x}= 0u(xwhere_{j},t_{n+1})=u(x_{j}-ak,t_{n})t. 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_{n+1}=t_{n+k}{Uabout^{n}}xand tracing characteristics back from_{j}(x._{j},t_{n+1})

- 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.

- 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.

- 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
matlabcodes as the basis for these experiments. Compare the 1-, 2- and infinity-norm of the numerical solution error at a given fixed time,Tfor various valueshwith Courant number,v, constant and for variousvwithhconstant. Repeat the experiments with discontinuous data.

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

- Use von Neumann stability analysis to find a stability restriction for the explicit scheme

U^{n+1}_{j}= U^{n}_{j}+ kD( U^{n}_{j+1}- 2 U^{n}_{j}+ U^{n}_{j-1})/h^{2}

for the heat equationu._{t}= D u_{xx}

- Find the numerical solution to Q.1 using both Lax-Wendroff and symmetrized (first order) upwinding and Beam-Warming schemes. Take
d,_{0}=1vand_{0}=0.5X=0.05.

- What happens if you choose
v = ak/h = 1for 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.)