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:
- Homogeneous boundary conditions, uN = 0.
- Linear extrapolating boundary conditions, uN = 2uN−1− uN−2.
- Quadratic extrapolating boundary conditions, uN = 3uN−1− 3uN−2 + uN−3.
- Homogeneous Neumann boundary conditions, uN = uN−1.
- Antisymmetric boundary conditions, uN = −uN−1.
- First-order upwinding convective boundary conditions,
- 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.