Scalable Algorithms for High-Order Approximations on Three-Dimensional Compact Stencils

TL;DR

This paper introduces a scalable parallel algorithm for high-order compact approximations of 3D Helmholtz equations, leveraging separation of variables and FFT, with broad applicability to related linear systems.

Contribution

The authors develop a general, efficient parallel direct solver for high-order compact schemes on 3D grids, extending previous methods and demonstrating high performance in various computing environments.

Findings

Achieves near-optimal complexity for 4th and 6th order schemes

Successfully implements in OpenMP, MPI, and hybrid environments

Demonstrates high efficiency on multicore and cluster systems

Abstract

This paper presents an efficient parallel direct algorithm with near-optimal complexity for the compact fourth and sixth-order approximation of the three-dimensional Helmholtz equations [1] with the problem coefficient depending on only one of the coordinate directions. The developed method is based on a combination of the separation of variables technique and a Fast Fourier Transform (FFT) type method. Similar direct solvers for the lower-order approximations of the two and three-dimensional Helmholtz equation were considered in several previous publications by the authors and other researchers (see e.g. [2,3,4,5]). The authors also consider a generalization of the presented algorithm to the solution of a wide class of linear systems obtained from approximation on the compact 27-point three-dimensional stencils on the rectangular grids with similar requirements on the stencil coefficients. The general restrictions on the coefficients in the considered class of compact schemes are developed and presented. This class includes the second, fourth and sixth-order compact approximation schemes for the three-dimensional Helmholtz equation considered in this paper and our previous publications [2,3,6]. As an example of the diversity of applications of the developed general method, the direct parallel implementation of a compact fourth-order approximation scheme for a convection-diffusion equation is considered. The results of the implementation of this method in OpenMP, MPI and hybrid programming environments on the multicore computers and multiple node clusters are presented and discussed. The results demonstrate the high efficiency of the proposed direct solvers for many important applications on the structured grid with the corresponding 27-diagonal matrices.