MAE290C- HOMEWORK 4 Solved

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, uN = 0.

2.   Linear extrapolating boundary conditions, uN = 2uN−1− uN−2.

3.   Quadratic extrapolating boundary conditions, uN = 3uN−1− 3uN−2 + uN−3.

4.   Homogeneous Neumann boundary conditions, uN = uN−1.

5.   Antisymmetric boundary conditions, uN = −uN−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.