A GPU-accelerated Cartesian grid method for PDEs on irregular domain

TL;DR

This paper introduces a GPU-accelerated Cartesian grid method for solving PDEs on irregular domains using the kernel-free boundary integral approach, significantly improving computational speed and efficiency over CPU implementations.

Contribution

The paper develops GPU-based algorithms for the KFBI method, enabling efficient single- and multi-GPU solutions for PDEs on irregular domains, with demonstrated high accuracy and speed.

Findings

Single-GPU solver is 50-200 times faster than CPU implementations.

Multi-GPU solver achieves up to 60% parallel efficiency.

The method is second-order accurate and computationally efficient.

Abstract

The kernel-free boundary integral (KFBI) method has successfully solved partial differential equations (PDEs) on irregular domains. Diverging from traditional boundary integral methods, the computation of boundary integrals in KFBI is executed through the resolution of equivalent simple interface problems on Cartesian grids, utilizing fast algorithms. While existing implementations of KFBI methods predominantly utilize CPU platforms, GPU architecture's superior computational capabilities and extensive memory bandwidth offer an efficient resolution to computational bottlenecks. This paper delineates the algorithms adapted for both single-GPU and multiple-GPU applications. On a single GPU, assigning individual threads can control correction, interpolation, and jump calculations. The algorithm is expanded to multiple GPUs to enhance the processing of larger-scale problems. The arrowhead decomposition method is employed in multiple-GPU settings, ensuring optimal computational efficiency and load balancing. Numerical examples show that the proposed algorithm is second-order accurate and efficient. Single-GPU solver speeds 50-200 times than traditional CPU while the eight GPUs distributed solver yields up to 60% parallel efficiency.