Abstract
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.
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Notes
- 1.
Note that in problems with a quadratic fast potential both the Deuflhard method and the impulse method are identical as suggested in [10].
- 2.
A symplectic integrator is a scheme that intends to simulate a Hamiltonian system numerically, while it preserves its underlying symplectic structure [12].
- 3.
One can use the MATLAB command sysd = c2d(sys,‘foh’) in order to discretize a linear-time-invariant system.
- 4.
Linearly implicit methods need only to solve a linear equation set, no matter the system we simulate is linear or nonlinear (see [14] for more details).
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Rahrovani, S., Abrahamsson, T., Modin, K. (2016). Stability Limitations in Simulation of Dynamical Systems with Multiple Time-Scales. In: Kerschen, G. (eds) Nonlinear Dynamics, Volume 1. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-15221-9_7
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DOI: https://doi.org/10.1007/978-3-319-15221-9_7
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