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