High order volume preserving integrators for three kinds of divergence-free vector fields via commutators
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In this paper, we focus on construction of high order volume preserving integrators for divergence-free vector fields of the monomial basis, exponential basis and tensor product of the monomial and exponential basis. We first prove that the commutators of elementary divergence-free vector fields (EDFVF) of these three kinds are still divergence-free vector fields of the same kind. For EDFVFs of these three kinds, we construct high order volume preserving integrators using the multi-commutators. Moreover, we consider ordering of EDFVFs and their commutators to reduce the error of the schemes, showing by numerical tests that the strategies in  work well.