Relaxation Exponential Runge-Kutta Methods and Their Applications to Semilinear Dissipative/Conservative Systems

This paper presents a family of novel relaxation exponential Runge-Kutta methods for semilinear partial differential equations with dissipative/conservative energy. The novel methods are developed by using the relaxation idea and adding a well-designed governing equation to explicit exponential Runge-Kutta methods. It is shown that the proposed methods can be of high-order accuracy and energy-stable/conserving with mild time step restrictions. In contrast, the previous explicit exponential-type methods are not energy-conserving. Several numerical experiments on KdV equations, Schrödinger equations and Navier-Stokes equations are carried out to illustrate the effectiveness and high efficiency of the methods.