Answer Happy

1. Solve the advection equation using the FTCS-and the Lax-Wendroff-scheme with periodic boundary conditions. Use one spatial dimension x and the time t as independent variables. Use a cosine modulated Gaussian pulse as inital value, i.e., a(x,t = 0) = cos(k(x-ct)) exp with k =, G=0.1, and c=1. Calculate and plot the solution a(x,t) as a mesh-plot (Matlab command mesh). Before you start: Which solution would you would expect? Using the FTCS-scheme: Why is the numerical solution not correct? Using the Lax-Wendroff scheme: Test at least three time steps t below, at, and above the maximum stable time step. Describe the effect on the solution. Amplitude versus x and t using LW for r< (stable) 0.5 NON 0.5 -(x-ct)2) 20² L -0.5 Position Position Amplitude versus x and t using LW for = (borderline stable) -0.5 Position Amplitude versus x and t using LW for r> (unstable) 0.5 Time -0.5 0.5 Time 0.5 Time
0.5 1015 0.5 E 2025 W Amplitude versus x and t using LW for r < (stable) -0.5 Position Time Amplitude versus x and t using LW for 7= (borderline stable) Position -0.5 Position Amplitude versus x and t using LW for > (unstable) -0.5 0.5 0 0.5 Time 0.5 Time