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

TL;DR

The paper introduces the N-shaped partition method, a parallel algorithm for solving tridiagonal systems in PDEs, optimized for GPU and MPI architectures, enabling large-scale simulations without memory constraints.

Contribution

It presents a novel parallel implementation of the Crank-Nicolson algorithm using the N-shaped partition method, suitable for large systems on GPU and HPC architectures.

Findings

Efficient GPU implementation for large tridiagonal systems.

Analytical estimation of optimal parameters for the algorithm.

Demonstrated scalability and suitability for large PDE problems.

Abstract

We develop an algorithm to solve tridiagonal systems of linear equations, which appear in implicit finite-difference schemes of partial differential equations (PDEs), being the time-dependent Schr\"{o}dinger equation (TDSE) an ideal candidate to benefit from it. Our N-shaped 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 well-suited 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 ab-initio studies might be otherwise prohibitive both because of large shared-memory requirements and computation times.