Multirate Exponential Rosenbrock Methods

In this paper we present a novel class of methods for high order accurate integration of multirate systems of ordinary differential equation initial-value problems. Following from recent work on multirate exponential Runge--Kutta (MERK) methods, that construct multirate schemes by approximating the action of matrix $\varphi$-functions within explicit exponential Runge--Kutta methods, the proposed methods similarly build off of explicit exponential Rosenbrock (ExpRB) methods. By leveraging the exponential Rosenbrock structure, the proposed Multirate Exponential Rosenbrock (MERB) methods consist of the solution to a sequence of modified ``fast'' initial-value problems, that may themselves be approximated through subcycling any desired IVP solver. In addition to proving how to construct MERB methods from certain classes of ExpRB methods, we provide rigorous convergence analysis of the resulting schemes, and present candidate MERB schemes of orders two through six. We then present numerical simulations to confirm these theoretical convergence rates, and to compare the efficiency of MERB methods against recently-introduced multirate MERK and MRI-GARK methods.