2.7 Eigenvalue Stability of Finite Difference Methods | 2.7 Eigenvalue Stability of Finite Difference Methods | Unit 2: Numerical Methods for Partial Differential Equations | Computational Methods in Aerospace Engineering | Aeronautics and Astronautics

2.7.4 Stability Exercises

Exercise 1

Download the MATLAB^® code in Section 2.7.2. Modify this code to compute the eigenvalues of the FTBS method with periodic boundary conditions. For CFL=1, where are there eigenvalues of \(A\)?

All in the left half plane

Two in the left half plane, the rest in a circle in the right half plane

Two in the right half plane, the rest in a circle in the right half plane

"Along the imaginary axis

Answer:

To implement a backward spatial discretization, we modify the matrix A:

for i = 2:Nx-1 A(i,i) = u/dx; A(i,i-1) = -u/dx; end

Exercise 2

Compute analytically the eigenvalues for the FTBS method with periodic boundary conditions. The eigenvalues are

\(-\frac{u}{\Delta x} -\frac{u}{\Delta x}\left[\cos \left(2\pi \frac{n}{N}\right) - i\sin \left(2\pi \frac{n}{N}\right)\right]\)

\(-\frac{u}{\Delta x} +\frac{u}{\Delta x}\left[\cos \left(2\pi \frac{n}{N}\right) - i\sin \left(2\pi \frac{n}{N}\right)\right]\)

\(\frac{u}{\Delta x} +\frac{u}{\Delta x}\left[\cos \left(2\pi \frac{n}{N}\right) + i\sin \left(2\pi \frac{n}{N}\right)\right]\)

\(-\frac{u}{\Delta x} +\frac{u}{\Delta x}\left[\cos \left(2\pi \frac{n}{N}\right) + i\sin \left(2\pi \frac{n}{N}\right)\right]\)

Answer: We use the formula for the eigenvalues of a circulant matrix with \(a_1= -u/\Delta x\) and \(a_ N = u/ \Delta x\).