Stability Limitations in Simulation of Dynamical Systems with Multiple Time-Scales

This paper focuses on the stability properties of a recently proposed exponential integrator particularly in simulation of highly oscillatory systems with multiple time-scales. The linear and nonlinear stability properties of the presented exponential integrator have been studied. We illustrate this with the Fermi–Pasta–Ulam (FPU) problem, a highly oscillatory nonlinear system known as a test benchmark for multi-scale time integrators. This example is also illustrative when studying the numerical resonance and algorithmic instability in the multi-time-stepping (MTS) methods, such as in exponential and/or trigonometric integration schemes, since it has no external input force and therefore no real physical resonance.