Fourth-order two-stage explicit exponential integrators for solving differential equations

TL;DR

This paper introduces two new explicit exponential Rosenbrock methods of fourth order with two stages, offering efficient and superconvergent solutions for large differential systems, outperforming traditional implicit schemes.

Contribution

The paper develops and proves convergence for the first explicit two-stage fourth-order exponential Rosenbrock schemes for large systems.

Findings

The new schemes are fully explicit and superconvergent.

Numerical experiments show improved efficiency over implicit methods.

The methods are suitable for large-scale differential equations.

Abstract

Among the family of fourth-order time integration schemes, the two-stage Gauss--Legendre method, which is an implicit Runge--Kutta method based on collocation, is the only superconvergent. The computational cost of this implicit scheme for large systems, however, is very high since it requires solving a nonlinear system at every step. Surprisingly, in this work we show that one can construct and prove convergence results for exponential methods of order four which use two stages only. Specifically, we derive two new fourth-order two-stage exponential Rosenbrock schemes for solving large systems of differential equations. Moreover, since the newly schemes are not only superconvergent but also fully explicit, they clearly offer great advantages over the two-stage Gauss--Legendre method as well as other time integration schemes. Numerical experiments are given to demonstrate the efficiency of the new integrators.