A GPU-Accelerated Cartesian Grid Method for the Heat, Wave and Schrödinger Equations on Irregular Domains

Based on Ying’s kernel-free boundary integral (KFBI) method [1], a second-order method for general elliptic partial differential equations (PDEs), this paper develops a GPU-accelerated KFBI method for the heat, wave and Schrödinger equations on the irregular domain. Since the limitation of time steps imposed by CFL conditions in the explicit scheme and the inadequate accuracy generated by the fully implicit scheme for the Laplacian operator, the paper selects a series of second-order time discrete schemes, and the Laplacian operator is split into explicit and implicit mixed ones. The Crank-Nicolson method is used to discretize the heat equation in temporal dimension while the implicit $θ$-scheme is for the wave equation. The Strang splitting method is applied to the Schrödinger equation. After discretizing the temporal dimension implicitly, the heat, wave and Schrödinger 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 instead of being discretized and corrected by the five-point difference method. A Cartesian grid-based KFBI method is used to solve the resulting elliptic equations. The KFBI method is accelerated by the graphics processing unit (GPU) with a parallel Cartesian grid solver, achieving a high degree of parallelism. Numerical results show that the proposed method has a second-order accuracy for the heat, wave, and Schrödinger equations. Additionally, the GPU-accelerated solvers for the three types of time-dependent equations are 30 times faster than CPU-based solvers.