High performance parallel algorithm for solving elliptic equations with non-separable variables

TL;DR

This paper introduces a highly efficient parallel algorithm for solving elliptic equations with non-separable variables, utilizing a differential Laplace operator preconditioner and parallel separation techniques, suitable for large-scale computations.

Contribution

It presents a novel parallel algorithm combining a differential Laplace preconditioner with a separation variable method for elliptic equations with non-separable variables.

Findings

Highly efficient for large processor counts

Effective in solving acoustic equations with integral Laguerre transformation

Utilizes parallel FFT and tridiagonal solvers for performance

Abstract

A parallel algorithm for computing the finite difference solution to the elliptic equations with non-separable variables is presented. The resultant matrix is symmetric positive definite, thus the preconditioning conjugate gradient or the chebyshev method can be applied. A differential analog to the Laplace operator is used as preconditioner. For inversion of the Laplace operator we implement a parallel version of the separation variable method, which includes the sequential FFT algorithm and the parallel solver for tridiagonal matrix equations (dichotomy algorithm). On an example of solving acoustic equations by the integral Laguerre transformation method, we show that the algorithm proposed is highly efficient for a large number of processors.