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<p>This paper mainly extends the discrete time waveform relaxation (DWR) method to stochastic differential equations (SDEs) and analyzes the stability of the method. Our research innovatively delves into the sufficient conditions for the stability of the DWR method applied to <i>d</i>-dimensional Itô SDEs. Because the choice of the splitting function will affect the stability of the DWR method, this paper mainly proposes the sufficient conditions for three kinds of splitting functions based on block Gauss-Jacobi (BGJ), block Gauss-Seidel (BGS), and eigenvalue (EV) splittings. It is demonstrated that, under certain conditions, the DWR methods utilizing EV splitting exhibit superior stability in comparison to the DWR methods based on BGJ and BGS splittings. Additionally, during the stability calculation process, the DWR method under the explicit Euler scheme shows superior computational efficiency compared to its implicit Euler scheme when achieving equivalent error tolerances. Finally, the obtained results are supported by numerical simulation experiments.</p>

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Stochastic Stability of Waveform Relaxation Methods Corresponding to Different Splitting Techniques

  • 期刊论文

  • 发表日期：

    2025年12月13日

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  • Xuan Xin,
  • Lanting Wei,
  • Longbin Wu,
  • Xiaohua Ding

摘要

This paper mainly extends the discrete time waveform relaxation (DWR) method to stochastic differential equations (SDEs) and analyzes the stability of the method. Our research innovatively delves into the sufficient conditions for the stability of the DWR method applied to d-dimensional Itô SDEs. Because the choice of the splitting function will affect the stability of the DWR method, this paper mainly proposes the sufficient conditions for three kinds of splitting functions based on block Gauss-Jacobi (BGJ), block Gauss-Seidel (BGS), and eigenvalue (EV) splittings. It is demonstrated that, under certain conditions, the DWR methods utilizing EV splitting exhibit superior stability in comparison to the DWR methods based on BGJ and BGS splittings. Additionally, during the stability calculation process, the DWR method under the explicit Euler scheme shows superior computational efficiency compared to its implicit Euler scheme when achieving equivalent error tolerances. Finally, the obtained results are supported by numerical simulation experiments.