Explicit high-order energy-preserving exponential time differencing method for nonlinear Hamiltonian PDEs

Abstract In this paper, an explicit and energy-preserving exponential time differencing method is developed for Hamiltonian PDEs whose vector field can be separated into linear and nonlinear parts. First, the original equation is transformed into an equivalent system with quadratic energy conservation law by the scalar auxiliary variable (SAV) approach. Then the exponential time difference Runge-Kutta (ETDRK) method is applied to the semi-discrete SAV reformulation which conserves semi-discrete quadratic energy conservation law. Different from the standard ETDRK method, an additional differential equation of the auxiliary variable is introduced by the SAV reformulation, therefore, special treatments are requisite to maintain the accuracy of the ETDRK method and thus obtain high-order scheme. Finally, the energy preservation is achieved by the projection technique which is completely explicit because of the quadratization of the energy. Numerical experiments show that the proposed method is more effective than other comparison methods.