The paper proposes a GPU-accelerated KFBI method for solving the heat, wave, and Schrödinger equations on irregular domains. The key highlights are:
The paper first discretizes the temporal dimension of the heat, wave, and Schrödinger equations using implicit schemes like the Crank-Nicolson method, implicit θ-scheme, and Strang splitting, respectively. This transforms the time-dependent PDEs into a sequence of elliptic equations at each time step.
The KFBI method is then used to efficiently solve the resulting elliptic equations on irregular domains. The KFBI method avoids directly representing Green's functions and instead solves an equivalent interface problem on a Cartesian grid using fast elliptic solvers like the FFT-based solver.
To further enhance computational efficiency, the KFBI method is parallelized on a GPU. The simple Cartesian grid structure enables high parallelism, with each grid node assigned to a dedicated GPU thread. The paper discusses various GPU parallelization strategies, such as using shared memory and atomic functions, to optimize memory access and achieve a 30x speedup over CPU-based solvers.
Numerical results demonstrate that the proposed GPU-accelerated KFBI method achieves second-order accuracy for the heat, wave, and Schrödinger equations on irregular domains.
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arxiv.org
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