Using Eigenvalues to Investigate Numerical Oscillations

In numerically solving linear partial differential equations, carefully formed matrices are powered up as the algorithm iterates the solution through time. Depending upon the eigenvalues, the solution either blows up to non-numerical values, stabilizes with bounded oscillations, or stabilizes free of oscillations. This talk presents comparisons of eigenvalue conditions through extended von Neumann stability analysis of standard numerical methods for foundational partial differential equations. Known lower eigenvalue bounds for stability are not optimal, so computational testing is necessary. We will compare several implementations to automate detection of solution behavior for testing theoretical conjectures. Along the way, we will pause to enjoy some surprisingly beautiful figures resulting from such analysis and computation.