High-order Lagrange multiplier schemes for general Hamiltonian PDEs

TL;DR

This paper develops high-order, energy-preserving numerical schemes for Hamiltonian PDEs using a novel Lagrange multiplier approach that maintains the original energy exactly, broadening applicability and ensuring computational efficiency.

Contribution

Introduces a Lagrange multiplier method for constructing high-order energy-preserving schemes for Hamiltonian PDEs without requiring energy boundedness.

Findings

Exact energy preservation at discrete level

Broad applicability without energy boundedness requirement

Computational cost comparable to existing methods

Abstract

In this paper, we introduce a Lagrange multiplier approach to construct linearly implicit energy-preserving schemes of arbitrary order for general Hamiltonian PDEs. Unlike the widely used auxiliary variable methods, this novel approach does not require the nonlinear part of the energy to be bounded from below, thereby offering broader applicability. Moreover, this approach preserves the original energy exactly at both the continuous and discrete levels, as opposed to a modified energy preserved by the auxiliary variable methods. Rigorous proofs are provided for the energy conservation and numerical accuracy of all derived schemes. The trade-off for these advantages is the need to solve a nonlinear algebraic equation to determine the Lagrange multiplier. Nevertheless, numerical experiments show that the associated computational cost is generally not dominant, indicating that the new schemes retain computational efficiency comparable to the auxiliary variable-based schemes. Numerical results demonstrate the efficiency, accuracy, and structure-preserving properties of the proposed schemes.