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

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