Fourth-order two-stage explicit exponential integrators for time-dependent PDEs

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 turn out to be very competitive compared to the two-stage Gauss-Legendre method as well as other fourth-order time integration schemes. Numerical experiments are given to demonstrate the efficiency of the new integrators. (C) 2016 IMACS. Published by Elsevier B.V. All rights reserved.