The Nshaped partition method: A novel parallel implementation of the Crank Nicolson algorithm
Abstract
We develop an algorithm to solve tridiagonal systems of linear equations, which appear in implicit finitedifference schemes of partial differential equations (PDEs), being the timedependent Schrödinger equation (TDSE) an ideal candidate to benefit from it. Our Nshaped partition method optimizes the implementation of the numerical calculation on parallel architectures, without memory size constraints. Specifically, we discuss the realization of our method on graphics processing units (GPUs) and the Message Passing Interface (MPI). In GPU implementations, our scheme is particularly advantageous for systems whose size exceeds the global memory of a single processor. Moreover, because of its lack of memory constraints and the generality of the algorithm, it is wellsuited for mixed architectures, typically available in large high performance computing (HPC) centers. We also provide an analytical estimation of the optimal parameters to implement our algorithm, and test numerically the suitability of our formula in a GPU implementation. Our method will be helpful to tackle problems which require large spatial grids for which abinitio studies might be otherwise prohibitive both because of large sharedmemory requirements and computation times.
 Publication:

Computer Physics Communications
 Pub Date:
 June 2023
 DOI:
 10.1016/j.cpc.2023.108713
 arXiv:
 arXiv:2205.00856
 Bibcode:
 2023CoPhC.28708713L
 Keywords:

 Parallel CrankNicolson;
 Tridiagonal parallel solver;
 High performance computing;
 Physics  Computational Physics
 EPrint:
 Computer Physics Communications 287 (2023) 108713