Conserving mass, momentum, and energy for the Benjamin-Bona-Mahony, Korteweg-de Vries, and nonlinear Schr\"odinger equations

TL;DR

This paper introduces high-order numerical methods that preserve invariants like mass, momentum, and energy for several PDEs, improving long-term simulation accuracy.

Contribution

The authors develop explicit Fourier Galerkin-based schemes that conserve multiple invariants for PDEs, a novel combination of high-order accuracy and conservation.

Findings

Methods conserve invariants up to numerical precision.

Conservation reduces long-term numerical error growth.

Applicable to multiple PDEs including NLS and KdV.

Abstract

We propose and study a class of arbitrarily high-order numerical discretizations that preserve multiple invariants and are essentially explicit (they do not require the solution of any large systems of algebraic equations). In space, we use Fourier Galerkin methods, while in time we use a combination of orthogonal projection and relaxation. We prove and numerically demonstrate the conservation properties of the method by applying it to the Benjamin-Bona-Mahony, Korteweg-de Vries, and nonlinear Schr\"odinger (NLS) PDEs as well as a hyperbolic approximation of NLS. For each of these equations, the proposed schemes conserve mass, momentum, and energy up to numerical precision. We show that this conservation leads to reduced growth of numerical errors for long-term simulations.