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

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