Second-order linear structure-preserving modified finite volume schemes for the regularized long-wave equation

TL;DR

This paper introduces second-order, energy-preserving finite volume schemes for the regularized long-wave equation, ensuring accuracy and efficiency through novel Hamiltonian transformations and linear-implicit methods.

Contribution

It proposes a new approach transforming the Hamiltonian into a quadratic form, leading to efficient, energy-conserving schemes that are both second-order accurate and practical for computations.

Findings

Schemes preserve energy at the discrete level.

Methods achieve second-order accuracy in time and space.

Numerical experiments confirm accuracy and energy preservation.

Abstract

In this paper, based on the weak form of the Hamiltonian formulation of the regularized long-wave equation and a novel approach of transforming the original Hamiltonian energy into a quadratic functional, a fully implicit and three linear-implicit energy conservation numerical schemes are respectively proposed. The resulting numerical schemes are proved theoretically to satisfy the energy conservation law in the discrete level. Moreover, these linear-implicit schemes are efficient in practical computation because only a linear system need to be solved at each time step. The proposed schemes are both second order accurate in time and space. Numerical experiments are presented to show all the proposed schemes have satisfactory performance in providing accurate solution and the remarkable energy-preserving property.