Consider the one-way wave equation ut + uz = 0 on the interval (-1,3) and for t> 0 with the following two sets of initial conditions: 1- if |2 <1, u(2,0) = (5) otherwise, and u(1,0) = e-52. (6) (a) Use the forward-time forward-space me: ,n+1 +-u + 0, k h with right-point boundary condition unti where IM = 3 to compute an approx- imation to the solution at several (up to 40) time steps. Use h = 0.02 and 1 = k/h = 0.8. Demonstrate numerically (plot the solution) the instability of the scheme and show that the instability appears sooner with the less smooth initial data.
(b) Comment on the localization of the onset of instability for initial data (5) and give an estimate of the expected growth rate of the instability per time step.
(c) Using the left boundary condition u(-1) = 0, write a stable scheme and compute the corresponding approximation for data sets (5) and (6). Plot the approximations at representative time steps. Use again h = 0.02 and 1 = k/h=0.8. =