Conserving mass, momentum, and energy for the Benjamin-Bona-Mahony, Korteweg-de Vries, and nonlinear Schrödinger equations
Hendrik Ranocha, David I. Ketcheson
TL;DR
This paper develops a class of high-order, largely explicit full discretizations that preserve mass, momentum, and energy for BBM, KdV, and NLS equations by combining Fourier Galerkin spatial discretization with a projection-relaxation time-stepping strategy. The method leverages invariant-preserving projection operators and scalar solves per time step to achieve fully discrete conservation up to machine precision, enabling substantially improved long-time accuracy compared with schemes that conserve fewer invariants. Numerical experiments on multi-soliton interactions and hyperbolic-NLS variants demonstrate robust invariant conservation, favorable error growth, and major efficiency gains relative to existing two-invariant methods. The approach is flexible (allowing arbitrary order and baseline integrators), but requires periodic boundaries and global (not local) conservation, with promising extensions to higher dimensions and related integrable lattices discussed as future work.
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-Mahoney, Korteweg-de Vries, and nonlinear Schrödinger (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.