Explicit volume-preserving splitting methods for divergence-free ODEs by tensor-product basis decompositions

A Zanna

doi:10.1093/imanum/drt070

Abstract

[Received on 11 September 2012; revised on 15 August 2013] We discuss the construction of volume-preserving splitting methods based on a tensor product of single-variable basis functions. The vector field is decomposed as the sum of elementary divergence-free vector fields (EDFVFs), each of them corresponding to a basis function. The theory is a generalization of the monomial basis approach introduced in Xue & Zanna (2013, BIT Numer. Math., 53, 265–281) and has the trigonometric splitting of Quispel & McLaren (2003, J. Comp. Phys., 186, 308–316) and the splitting in shears of McLachlan & Quispel (2004, BIT, 44, 515–538) as special cases. We introduce the concept of diagonalizable EDFVFs and identify the solvable ones as those corresponding to the monomial basis and the exponential basis. In addition to giving a unifying view of some types of volume-preserving splitting methods already known in the literature, the present approach allows us to give a closed-form solution also to other types of vector fields that could not be treated before, namely those corresponding to the mixed tensor product of monomial and exponential (including trigonometric) basis functions.

Highlights

Volume preservation is an important property shared by several dynamical systems
We have presented a framework for generating volume-preserving splitting methods based on a decomposition of the divergence of the vector field using an appropriate tensor product of basis functions of one variable
The vector field is decomposed into the sum of elementary divergence-free vector fields (EDFVFs), each of them corresponding to a basis function

Summary

Introduction

Volume preservation is an important property shared by several dynamical systems. The principal example is the velocity field for an incompressible fluid flow and applications include the study of turbulence and of mixing of flows (see Arnold & Khesin, 1998). In Xue & Zanna (2013), we presented a novel approach for arbitrary polynomial divergence-free vector fields, by expanding the divergence equation in terms of the monomial basis, that allowed us to develop explicit volume-preserving splitting methods. Each xi obeys a differential equation with variable coefficients, indicating that one might find classes of diagonalizable EDFVFs whose solution can be given in closed form For this purpose, we study the behaviour of the basis function φj(x) as a function of time. When ci = 1, the equation becomes Φ = kiΦ, which has solution Φ(z) = li exp(kiz), where li is another constant of integration This corresponds to the basis function φji (xi) = likiekixi and it is useful to choose liki = 1. We might increase the order by using the commutators

Connection with the splitting in shears of Quispel and McLachlan

Trigonometric polynomials and Fourier bases

The method of expansion in a Fourier series

Tensor product of Fourier series and the monomial basis

Avoiding complex arithmetic

Concluding remarks