  In this paper we present a methodology for data accesses when solving batches
of Tridiagonal and Pentadiagonal matrices that all share the same
left-hand-side (LHS) matrix. The intended application is to the numerical
solution of Partial Differential Equations via the finite-difference method,
although the methodology is applicable more broadly. By only storing one copy
of this matrix, a significant reduction in storage overheads is obtained,
together with a corresponding decrease in compute time. Taken together, these
two performance enhancements lead to an overall more efficient implementation
over the current state of the art algorithms cuThomasBatch and cuPentBatch,
allowing for a greater number of systems to be solved on a single GPU. We
demonstrate the methodology in the case of the Diffusion Equation,
Hyperdiffusion Equation, and the Cahn--Hilliard Equation, all in one spatial
dimension. In this last example, we demonstrate how the method can be used to
perform $2^{20}$ independent simulations of phase separation in one dimension.
In this way, we build up a robust statistical description of the coarsening
phenomenon which is the defining behavior of phase separation. We anticipate
that the method will be of further use in other similar contexts requiring
statistical simulation of physical systems.
