The N-shaped partition method: A novel parallel implementation of the Crank Nicolson algorithm

We develop an algorithm to solve tridiagonal systems of linear equations, which appear in implicit finite-difference schemes partial differential equations (PDEs), being the time-dependent Schr\"{o}dinger equation (TDSE) ideal candidate benefit from it. Our N-shaped partition method optimizes implementation numerical calculation on parallel architectures, without memory size constraints. Specifically, we discuss realization our graphics processing units (GPUs) and Message Passing Interface (MPI). In GPU implementations, scheme is particularly advantageous for whose exceeds global a single processor. Moreover, because its lack constraints generality algorithm, it well-suited mixed typically available large high performance computing (HPC) centers. also provide analytical estimation optimal parameters implement test numerically suitability formula implementation. will be helpful tackle problems require spatial grids ab-initio studies might otherwise prohibitive both shared-memory requirements computation times.