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<p>In statistical mechanics and its applications, stochastic differential equations (SDEs) are crucial for modeling complex systems. This study proposes a parallel waveform relaxation (WR) framework for high-dimensional SDEs, based on the stochastic Runge-Kutta (SRK) method. The scheme, designed for parallel computing, enhances computational efficiency, particularly in applications such as the simulation of stochastic particle dynamics. Integrating the SRK method with a limit-based iterative scheme ensures convergence and high-order convergence for some SDEs. A rigorous analysis confirms the method’s reliability and effectiveness. Numerical experiments on linear and non-linear high-dimensional SDEs show the proposed scheme outperforms the original SRK method in both convergence rate and computational time. These results are beneficial for applications in statistical mechanics and reduce the computational load in large-scale system analysis.</p>

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Efficient Numerical Simulation for High-Dimensional Stochastic Systems: a Parallel Waveform Relaxation Approach Based on Stochastic Runge-Kutta Methods

  • 期刊论文

  • 发表日期：

    2026年2月9日

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  • Xuan Xin,
  • Zhenyu Wang,
  • Xiaohua Ding

摘要

In statistical mechanics and its applications, stochastic differential equations (SDEs) are crucial for modeling complex systems. This study proposes a parallel waveform relaxation (WR) framework for high-dimensional SDEs, based on the stochastic Runge-Kutta (SRK) method. The scheme, designed for parallel computing, enhances computational efficiency, particularly in applications such as the simulation of stochastic particle dynamics. Integrating the SRK method with a limit-based iterative scheme ensures convergence and high-order convergence for some SDEs. A rigorous analysis confirms the method’s reliability and effectiveness. Numerical experiments on linear and non-linear high-dimensional SDEs show the proposed scheme outperforms the original SRK method in both convergence rate and computational time. These results are beneficial for applications in statistical mechanics and reduce the computational load in large-scale system analysis.