Arbitrary high-order structure-preserving schemes for the generalized Rosenau-type equation

TL;DR

This paper develops high-order numerical schemes that preserve momentum and energy for the generalized Rosenau-type equation, combining symplectic Runge-Kutta methods with Fourier spectral techniques.

Contribution

It introduces a novel class of high-order, structure-preserving schemes for the Rosenau equation using symplectic Runge-Kutta and auxiliary variable approaches.

Findings

Schemes effectively preserve momentum and energy.

Numerical tests confirm high accuracy and stability.

Performance comparisons demonstrate advantages over existing methods.

Abstract

In this paper, we are concerned with arbitrarily high-order momentum-preserving and energy-preserving schemes for solving the generalized Rosenau-type equation, respectively. The derivation of the momentum-preserving schemes is made within the symplectic Runge-Kutta method, coupled with the standard Fourier pseudo-spectral method in space. Then, combined with the quadratic auxiliary variable approach and the symplectic Runge-Kutta method, together with the standard Fourier pseudo-spectral method, we present a class of high-order mass- and energy-preserving schemes for the Rosenau equation. Finally, extensive numerical tests and comparisons are also addressed to illustrate the performance of the proposed schemes.