  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.
