Arbitrary high-order linearly implicit energy-conserving schemes for the Rosenau-type equation

Abstract

In this paper, we present a class of high-order linearly implicit energy-conserving schemes for solving the generalized Rosenau-type equation. We firstly reformulate the equation into a skew-adjoint system with the energy conservation law. Then the nonlinear terms of the system are approximated with an extrapolation/prediction–correction technique and then a linearized energy-conserving system is obtained. Finally, the symplectic Runge–Kutta method in time and the Fourier pseudo-spectral method in space are employed to discretize the resulting linearized system and a fully discrete scheme is obtained, which is linearly implicit, high-order and energy-conserving. Numerical results are addressed to demonstrate the accuracy and efficiency of the proposed scheme.