Local energy-preserving scalar auxiliary variable approaches for general multi-symplectic Hamiltonian PDEs

Abstract

We develop two classes of general-purpose second-order integrators for the general multi-symplectic Hamiltonian system by incorporating a scalar auxiliary variable. Unlike the previous methods introduced in [22], [31], these new approaches do not impose constraints on the state function of multi-symplectic system, and can preserve the original local/global energy conservation laws exactly. Moreover, the approaches are computationally efficient, as they only require solving linear equations with the same constant coefficients at each time step along with some additional scalar nonlinear equations. We employ the proposed methods to solve various equations, and the numerical results validate their solution accuracy, effectiveness, robustness, and energy-preserving ability.