[SOLVED] MAE 290C HOMEWORK 4

Integrate numerically the linear wave equation

∂_tu + c∂_xu = 0,

in the domain 0 ≤ x < 10 with homogeneous initial conditions and u(x = 0,t) = sin(At) Solve using secondorder centered finite difference schemes with N = 200 grid points, ∆t = 0.01 and the following boundary conditions at the artificial exit:

1. Homogeneous boundary conditions, u_N = 0.
2. Linear extrapolating boundary conditions, u_N = 2u_N−1− u_N−2.
3. Quadratic extrapolating boundary conditions, u_N = 3u_N−1− 3u_N−2 + u_N−3.
4. Homogeneous Neumann boundary conditions, u_N = u_N−1.
5. Antisymmetric boundary conditions, u_N = −u_N−1.
6. First-order upwinding convective boundary conditions,

.

7. Second-order upwinding convective boundary conditions,

.

Perform an analytical study of the reflection of waves generated by the each scheme at the artificial boundary and discuss the results obtained for A = 0.1 and A = 3. Compare your numerical results with the analysis of each boundary condition.