Arbitrary high-order linearly implicit energy-preserving algorithms for Hamiltonian PDEs

TL;DR

This paper introduces a new class of high-order, linearly implicit energy-preserving algorithms for Hamiltonian PDEs using the exponential scalar auxiliary variable approach, enabling explicit, efficient, and accurate numerical solutions.

Contribution

It develops a systematic, high-order, energy-preserving scheme based on ESAV and symplectic Runge-Kutta methods, improving efficiency and applicability over traditional SAV schemes.

Findings

The schemes are energy-preserving and highly accurate.

They decouple solution variables and auxiliary variables, enhancing efficiency.

Numerical experiments confirm the schemes' effectiveness for Hamiltonian PDEs.

Abstract

In this paper, we present a novel strategy to systematically construct linearly implicit energy-preserving schemes with arbitrary order of accuracy for Hamiltonian PDEs. Such novel strategy is based on the newly developed exponential scalar variable (ESAV) approach that can remove the bounded-from-blew restriction of nonlinear terms in the Hamiltonian functional and provides a totally explicit discretization of the auxiliary variable without computing extra inner products, which make it more effective and applicable than the traditional scalar auxiliary variable (SAV) approach. To achieve arbitrary high-order accuracy and energy preservation, we utilize the symplectic Runge-Kutta method for both solution variables and the auxiliary variable, where the values of internal stages in nonlinear terms are explicitly derived via an extrapolation from numerical solutions already obtained in the preceding calculation. A prediction-correction strategy is proposed to further improve the accuracy. Fourier pseudo-spectral method is then employed to obtain fully discrete schemes. Compared with the SAV schemes, the solution variables and the auxiliary variable in these ESAV schemes are now decoupled. Moreover, when the linear terms are of constant coefficients, the solution variables can be explicitly solved by using the fast Fourier transform. Numerical experiments are carried out for three Hamiltonian PDEs to demonstrate the efficiency and conservation of the ESAV schemes.