[Solved] M232 Homework 9-Numerical Analysis II

Consider the Lax-Wendroff method

In Matlab, we might be tempted to implement this method with periodic boundary conditions on the grid x_i = ih for i = 1,2, ,m + 1 with h = (b a)/(m + 1) within a time-stepping loop as

U(1) = U(1) 0.5 nu (U(2) U(m + 1))

+0.5 nu^2 (U(m + 1) 2.0 U(1) + U(2));

U(2 : m) = U(2 : m) 0.5 nu (U(3 : m + 1) U(1 : m 1))

+0.5 nu^2 (U(1 : m 1) 2.0 U(2 : m) + U(3 : m + 1));

U(m + 1) = U(m + 1) 0.5 nu (U(1) U(m))

+0.5 nu^2 (U(m) 2.0 U(m + 1) + U(1));

Note that periodic boundary conditions mean that U₀ = U_m+1, so we do not solve for U₀ in the equations above. However, this implementation is incorrect. Can you find the bug?

2. Consider the numerical scheme:
   • With which system of PDEs is this scheme consistent?

• Find the modified equations for this scheme.
• Using von Neumann stability analysis find conditions on k and h sufficient for this scheme to be stable.

3. The m-file advection LW pbc.m implements the Lax-Wendroff method for the advection equation on 0 x 1 with periodic boundary conditions.
   • Observe how this method behaves with m + 1 = 50,100,200 grid points. Change the final time to tfinal = 0.1 and use the m-files error table.m and error loglog.m to verify second order accuracy.
   • Modify the m-file to create a version advection up pbc.m implementing the upwind method and verify that this is first order accurate.
   • Keep m fixed and observe what happens with advection up pbc.m if the time step k is reduced, e.g. try k = 0.4h, k = 0.2h and k = 0.1h. When a convergent method is applied to an ODE we expect better accuracy as the time step is reduced and we can view the upwind method as an ODE solver applied to a Method of Lines system. However, you should observe decreased accuracy as k 0 with h Explain this apparent paradox. (Hint: What ODE system are we solving with more accuracy? You might also consider the modified equation given by (10.44) in LeVeque).

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