Papers › A new class of high-order methods for multirate differential equations
A new class of high-order methods for multirate differential equations
Vu Thai Luan, Rujeko Chinomona, Daniel R. Reynolds
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This work focuses on the development of a new class of high-order accurate methods for multirate time integration of systems of ordinary differential equations. The proposed methods are based on a specific subset of explicit one-step exponential integrators. More precisely, starting from an explicit exponential Runge--Kutta method of the appropriate form, we derive a multirate algorithm to approximate the action of the matrix exponential through the definition of modified "fast" initial-value problems. These fast problems may be solved using any viable solver, enabling multirate simulations through use of a subcycled method. Due to this structure, we name these Multirate Exponential Runge--Kutta (MERK) methods. In addition to showing how MERK methods may be derived, we provide rigorous convergence analysis, showing that for an overall method of order p, the fast problems corresponding to internal stages may be solved using a method of order p-1, while the final fast problem corresponding to the time-evolved solution must use a method of order p. Numerical simulations are then provided to demonstrate the convergence and efficiency of MERK methods with orders three through five on a series of multirate test problems.
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