Consider the problem

<jsm>u_t+au_{x}=0</jsm>, <jsm> t\ge0,</jsm> and <jsm>-\infty <x< \infty</jsm>

with initial condition

<jsm>u(x,0)=g(x).</jsm>

Task 1 Implement the explicit difference formulae of chapter 7.3 in the note. In particular (7.11) and (7.12) verify that (7.12) is always unstable while (7.11) converges.

Task 2 Implement Lax-Wendroff and Leap-Frog and verify numerically their order in space and time.

Task 3 Consider finally the inviscid Burgers equation

<jsm>u_t+\left(\frac{1}{2}u^2\right)_x =0</jsm>

discretize the time derivative and the space derivative in this equation using the same approximation as in formula (7.11). Verify numerically that the method converges.