Stability analysis of the numerical Method of characteristics applied to a class of energy-preserving systems. Part II: Nonreflecting boundary conditions
For researchers using the Method of characteristics, this clarifies when von Neumann instability does not imply practical instability, potentially expanding the range of usable schemes.
The paper shows that non-periodic boundary conditions can stabilize numerical schemes that are unstable under periodic boundary conditions, contradicting a common belief. It explains the mechanism and identifies cases where boundary conditions do not affect stability.
We show that imposition of non-periodic, in place of periodic, boundary conditions (BC) can alter stability of modes in the Method of characteristics (MoC) employing certain ordinary-differential equation (ODE) numerical solvers. Thus, using non-periodic BC may render some of the MoC schemes stable for most practical computations, even though they are unstable for periodic BC. This fact contradicts a statement, found in some literature, that an instability detected by the von Neumann analysis for a given numerical scheme implies an instability of that scheme with arbitrary (i.e., non-periodic) BC. We explain the mechanism behind this contradiction. We also show that, and explain why, for the MoC employing some other ODE solvers, stability of the modes may be unaffected by the BC.