High order volume preserving integrators for three kinds of divergence-free vector fields via commutator

TL;DR

This paper develops high order volume-preserving integrators for divergence-free vector fields using commutator methods, applicable to monomial, exponential, and tensor product bases, with numerical validation of error reduction strategies.

Contribution

It introduces a novel approach to construct high order volume-preserving integrators for divergence-free fields via multi-commutators, expanding the applicability to three basis types.

Findings

Commutators of elementary divergence-free vector fields remain divergence-free.

Constructed integrators achieve high order accuracy.

Numerical tests confirm effective error reduction strategies.

Abstract

In this paper, we focus on the construction of high order volume preserving in- tegrators for divergence-free vector fields: the monomial basis, the exponential basis and tensor product of the monomial and the exponential basis. We first prove that the commutators of elementary divergence-free vector fields (EDFVF) of those three kinds are still divergence-free vector fields of the same kind. Assuming then there is only diagonal part of divergence-free vector field of the monomial basis, for those three kinds of divergence-free vector fields, we construct high order volume-preserving inte- grators using the multi-commutators for EDFVFs. Moreover, we consider the ordering of the EDFVFs and their commutators to reduce the error of the schemes, showing by numerical tests that the strategy in [9] works very well.