2. (a) Solve the linear equation for t = 1 with h = 0.01 and CFL number v=k/h = 0.5, U + u = 0 u(x, 0) = 0, x < 0; u(x, 0) = 1, x 0, using the leap-frog scheme and the Lax-Wendroff scheme. (b) Solve the equation Ut+Uxxx = 0, u(x, 0) = 0, x < 0; u(x, 0) = 1, x 0, analytically in terms of the Airy function. (c) Compare the results of (a) and (b) with rescaling. Find the order of both schemes around discontinuity. 3. Use the nonconservative upwind scheme to solve the Riemann problem. u +ux = 0. u(x, 0) = u, x < 0; u(x, 0) = ur, x 0. (a) What is the exact shock speed for u = 1 and 0 < u, < 1? What is the corresponding numerical shock speed as the CFL number approaches 0? (b) Do the same problem for the equation u++uu = 0. 4. Find the numerical solution at t = T to the nonlinear Riemann problem. u+f(u) = 0, f(u) = u/2. u(x, 0) = u, x