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

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.