This paper introduces a second-order method for solving general elliptic partial differential equations (PDEs) on irregular domains using GPU acceleration, based on Ying's kernel-free boundary integral (KFBI) method. The method addresses limitations imposed by CFL conditions in explicit schemes and accuracy issues in fully implicit schemes for the Laplacian operator. To overcome these challenges, the paper employs a series of second-order time discrete schemes and splits the Laplacian operator into explicit and implicit components. Specifically, the Crank-Nicolson method discretizes the heat equation in the temporal dimension, while the implicit scheme is used for the wave equation. The Schrodinger equation is treated using the Strang splitting method. By discretizing the temporal dimension implicitly, the heat, wave, and Schrodinger equations are transformed into a sequence of elliptic equations. The Laplacian operator on the right-hand side of the elliptic equation is obtained from the numerical scheme rather than being discretized and corrected by the five-point difference method. A Cartesian grid-based KFBI method is employed to solve the resulting elliptic equations. GPU acceleration, achieved through a parallel Cartesian grid solver, enhances the computational efficiency by exploiting high degrees of parallelism. Numerical results demonstrate that the proposed method achieves second-order accuracy for the heat, wave, and Schrodinger equations. Furthermore, the GPU-accelerated solvers for the three types of time-dependent equations exhibit a speedup of 30 times compared to CPU-based solvers.

翻译：本文提出了一种基于Ying的无核边界积分（KFBI）方法，利用GPU加速求解不规则区域上一般椭圆型偏微分方程（PDEs）的二阶方法。该方法克服了显式格式中CFL条件对拉普拉斯算子的限制以及全隐式格式的精度问题。为应对这些挑战，本文采用了一系列二阶时间离散格式，并将拉普拉斯算子分裂为显式和隐式分量。具体而言，Crank-Nicolson方法用于热方程的时间离散，而隐式格式则用于波动方程。薛定谔方程采用Strang分裂方法处理。通过隐式离散时间维度，热方程、波动方程和薛定谔方程被转化为一系列椭圆方程。椭圆方程右侧的拉普拉斯算子由数值格式直接获得，而非通过五点差分法离散并校正。采用基于笛卡尔网格的KFBI方法求解生成的椭圆方程。通过并行笛卡尔网格求解器实现的GPU加速，利用高度并行性提升了计算效率。数值结果表明，该方法对热方程、波动方程和薛定谔方程均实现了二阶精度。此外，与基于CPU的求解器相比，针对三类时间依赖方程的GPU加速求解器实现了30倍的加速比。