Eigenvalues and Timestep restriction

Question

For an equation of the kind $u_t=Au$, time step is usually defined by ensuring that the eigenvalues are within the stability region of the time-stepping scheme employed.

1. If the eigenvalues are on positive real axis, then does that mean that the system is unconditionally unstable for all time stepping methods?
2. Is it possible to get a scheme that is stable in one time-stepping scheme but unstable in a different time stepping scheme(no matter how small the time step for the other scheme?). My feeling is that if the scheme is proven unstable for one time-stepping scheme, no matter how small the time-step (say, Forward Euler), then it is unstable for all time-stepping schemes because other TVD RK methods and multi-step methods are just convex combinations of the forward Euler method. Am I correct in assuming that? On the other hand, if it is stable in one time-stepping scheme, does it imply that it is stable in all other schemes by suitably adjusting the time-step?
3. If I use, for example, Discontinuous Galerkin methods for equations like shallow water, then the matrix A changes for every time step, does one calculate the eigenvalues at every time-step to ensure stability?

Answer 1

If the eigenvalues are on positive real axis, then does that mean that the system is unconditionally unstable for all time stepping methods?

No. If you take a large enough time step with backward Euler, then the scheme will be stable, but probably inaccurate. The underlying continuous problem is unstable (in the sense of dynamical systems).

Is it possible to get a scheme that is stable in one time-stepping scheme but unstable in a different time stepping scheme(no matter how small the time step for the other scheme?). My feeling is that if the scheme is proven unstable for one time-stepping scheme, no matter how small the time-step (say, Forward Euler), then it is unstable for all time-stepping schemes because other TVD RK methods and multi-step methods are just convex combinations of the forward Euler method. Am I correct in assuming that? On the other hand, if it is stable in one time-stepping scheme, does it imply that it is stable in all other schemes by suitably adjusting the time-step?

Take your system with positive eigenvalues. It's stable for backward Euler for sufficiently large time steps, and unstable when using forward Euler or the trapezoidal rule with any time step. Again, accuracy might be an issue when using backward Euler with large time steps.

If I use, for example, Discontinuous Galerkin methods for equations like shallow water, then the matrix A changes for every time step, does one calculate the eigenvalues at every time-step to ensure stability?

Not usually, no. Adaptive time steppers (for instance, SUNDIALS, which implements BDF methods and Adams-Bashforth methods) will use heuristics to detect both stability and accuracy issues to adjust time steps accordingly. When using a fixed time step, instabilities can sometimes be detected by eye (preferably via plotting; potentially via the raw numerical results in obvious cases, or if you have good intuition). Extremal eigenvalues could be estimated, for instance, via GMRES when doing each linear solve, which would be used to confirm the suspected diagnosis of instability. (PETSc is a good choice of package for such diagnostics.) Another test would be to decrease the time step and compare results via a convergence study; such a strategy isn't foolproof, but often works well as a diagnostic.