An assessment of component matrix reordering in sequential exponential splitting procedures

Exponential splitting plays an increasingly important role in scientific computations. The strategy targets the matrix within a matrix exponential operator, which may be generated by a semi-discretization scheme. This matrix is then split into a collection of relatively simple, but non-commuting, component matrices. Based on these, the original exponential operator is decomposed into a combination of sub-matrix exponentials to approximate solutions of underlying linear or nonlinear partial differential equations. Sequential splitting is a benchmark method in the literature. But a natural question arises: what are possible consequences for using different matrix orderings in a sequential splitting? Our study indicates that the impact of reordering non-commuting component matrices is limited. The theoretical analysis is further supported by numerical experiments based on both randomly generated matrices with realistic application characteristics and test cases simulating fuel combustion. This study also sheds light on related analyses of more general exponential splitting formulas.