Explicit Volume-Preserving Splitting Methods for Polynomial Divergence-Free Vector Fields

TL;DR

This paper introduces explicit volume-preserving splitting methods for polynomial divergence-free vector fields of any degree, enabling exact integration of split fields and resulting in efficient, high-quality numerical solutions.

Contribution

It develops a novel decomposition approach using monomial basis functions to create explicit, volume-preserving integrators for polynomial divergence-free vector fields.

Findings

Methods outperform standard integrators in solution quality

Methods are computationally efficient

Exact integration of split vector fields is achieved

Abstract

We present new, explicit, volume-preserving vector fields for polynomial divergence-free vector fields of arbitrary degree (both positive and negative). The main idea is to decompose the divergence polynomial by means of an appropriate basis for polynomials: the monomial basis. For each monomial basis function, the split fields are then identified by collecting the appropriate terms in the vector field so that each split vector field is divergence free. We show that each split field can be integrated exactly by analytical methods. Thus, the composition yields a volume preserving numerical method. Our numerical tests indicate that the methods compare favorably to standard integrators both in the quality of the numerical solution and the computational effort.