Software design for finite difference schemes based on index notation

Abstract

A formulation of finite difference schemes based on the index notation of tensor algebra is advocated. Finite difference operators on regular grids may be described as sparse, banded, “tensors”. Especially for higher space dimensions, it is claimed that a band tensor formulation better corresponds to the inherent problem structure than does conventional matrix notation. Tensor algebra is commonly expressed using index notation. The standard index notation is extended with the notion of index offsets, thereby allowing the common traversal of band tensor diagonals. The transition from mathematical index notation to implementation is presented. It is emphasized that efficient band tensor computations must exploit the particular problem structure, which calls for a combination of general index notation software with special-purpose band tensor routines.