Constructing invariant-preserving numerical schemes based on Poisson and Nambu brackets

Abstract

A novel procedure for designing invariant-preserving numerical schemes for Poisson and Nambu systems is proposed. Such systems include important physical problems such as the Korteweg--de Vries equation and the shallow water equations, where often some physical invariants control the dynamics of the solutions. By the new procedure, numerical schemes that preserve one or two such invariants can be constructed. The key is a clever discretization of the brackets. Numerical results for the shallow water equations using a fully discretized extension of the legendary Arakawa--Lamb scheme confirm the validity of our procedure.