Efficient simulation of a slow-fast dynamical system using multirate finite difference schemes

Abstract

We consider a system of ordinary differential equations describing a slow-fast dynamical system, in particular, a predator-prey system that is highly susceptible to local time variations. This model exhibits coexistence of predator- prey dynamics in the case when the prey population grows much faster than that of the predators with a quite diversied time response. For particular parametric values their interactions show a stable relaxation oscillation in the positive octant. Such characteristics are difficult to mimic using conventional time integrators that are used to solve systems of ordinary differential equations. To resolve this, we design and analyze multirate time integration methods to solve a mathematical model for a slow- fast dynamical system. Proposed methods are based on using extrapolation multirate discretisation algorithms. Through these methods, we reduce the integration time by integrating the slow sub-system with a larger step length than the fast sub-system. This allows us to efficiently solve multiscale ordinary differential equations. Besides theoretical results, we provide thorough numerical experiments which conrm that these multirate schemes outperform corresponding single-rate schemes substantially
both in terms of computational work and CPU times.

Mathematics Subject Classication (2010): 34C26, 65L05, 65L11, 65L12, 65L20.

Key words: Slow-fast dynamical systems, oscillations, multirate schemes, convergence analysis.