Explicit and CPU/GPU parallel energy-preserving schemes for the Klein-Gordon-Schr\"odinger equations

TL;DR

This paper introduces a dual-partition AVF method for Klein-Gordon-Schrödinger equations that is energy-preserving, explicitly decoupled, and highly efficient for CPU and GPU parallel computing, especially in high dimensions.

Contribution

It extends the AVF method to multivariable coupled systems, creating a fully explicit, energy-preserving scheme with pointwise decoupling suitable for parallel implementation.

Findings

The scheme conserves energy accurately in numerical tests.

It achieves computational complexity of O(N^d) per time step.

Parallel implementation on CPUs and GPUs significantly accelerates high-dimensional problem solving.

Abstract

A highly efficient energy-preserving scheme for univariate conservative or dissipative systems was recently proposed in [Comput. Methods Appl. Mech. Engrg. 425 (2024) 116938]. This scheme is based on a grid-point partitioned averaged vector field (AVF) method, allowing for pointwise decoupling and easy implementation of CPU parallel computing. In this article, we further extend this idea to multivariable coupled systems and propose a dual-partition AVF method that employs a dual partitioning strategy based on both variables and grid points. The resulting scheme is decoupled, energy-preserving, and exhibits greater flexibility. For the Klein-Gordon-Schr\"odinger equations, we apply the dual-partition AVF method and construct fully explicit energy-preserving schemes with pointwise decoupling, where the computational complexity per time step is $\mathcal{O}(N^d)$, with $d$ representing the problem dimension and $N$ representing the number of grid points in each direction. These schemes not only enable CPU parallelism but also support parallel computing on GPUs by adopting an update strategy based on a checkerboard grid pattern, significantly improving the efficiency of solving high-dimensional problems. Numerical experiments confirm the conservation properties and high efficiency of the proposed schemes.