Explicit and CPU/GPU parallel energy-preserving schemes for the Klein-Gordon-Schrödinger equations

TL;DR

This work introduces a dual-partition averaged vector field (DP-AVF) approach that combines variable-based and grid-point-based partitioning to produce energy-preserving, decoupled, and explicitly computable time integrators for multivariable Hamiltonian systems. Applied to the Klein-Gordon-Schrödinger equations, the DP-AVF framework yields fully explicit, energy-conserving schemes with pointwise decoupling and per-step cost $\mathcal{O}(N^d)$, enabling efficient CPU and GPU parallelization via a checkerboard update pattern. The paper develops both the theoretical DP-AVF construction and practical explicit implementations, including a second-order DP-AVF2 scheme via symmetric composition with its adjoint. Extensive 2D and 3D numerical experiments confirm exact discrete energy conservation, high accuracy, and substantial parallel speedups, highlighting the method’s potential for large-scale, high-dimensional simulations in physics and engineering.

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-Schrödinger 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.

Figures (14)

• Figure 1: Illustration of two orderings for defining the discrete gradient. The blue circles represent the unknowns, and the dashed arrows indicate the computing orders. (a) Each grid point follows the ordering \ref{['order1']}; (b) Each grid point follows the ordering \ref{['order2']}.
• Figure 2: A general update order on a $4 \times 4$ grid (left). The arrows indicate the update order for each grid point. Four representative grid points are selected to illustrate the detailed implementation, with their corresponding stencils shown on the right. Green nodes represent known values at time level $n$, while blue nodes represent unknown values at time level $n+1$. Rectangular nodes incorporate the periodic boundary condition.
• Figure 3: Parallelizable update order of the base method (left) and adjoint method (right) for the KGS equations in 2D.
• Figure 4: Parallelizable update order of the base method (left) and adjoint method (right) for the KGS equations in 3D.
• Figure 5: 2D checkerboard update order for GPU parallelization of the DP-AVF2 scheme for the KGS equations.

Theorems & Definitions (12)

• Theorem 2.1
• proof
• Remark 3.1
• Lemma 3.1
• Proof
• Theorem 3.1
• Proof
• Remark 3.2
• Remark 3.3
• Theorem 3.2