Scholarlink

<p>It is well known that exponential Runge-Kutta methods are widely used for solving semilinear <i>stiff</i> parabolic equations, where exponential elliptic operators are often encountered and the matrix exponential functions are inevitable. To develop a probabilistic approach for solving semilinear <i>stiff</i> parabolic equations without using the matrix exponential functions, in this paper, we give a probabilistic representation for a class of exponential elliptic operators under the periodic boundary condition via an expectation operator w.r.t. a diffusion process. Such a representation is a natural result of the nonlinear Feynman-Kac formula, from which we can write the solutions of a class of semilinear parabolic equations in forms of integral equation characterized by the expectation operator. Based on this integral equation, we propose a class of explicit time semidiscrete stochastic Runge-Kutta (SRK) methods for a class of semilinear parabolic equations, derive their <i>stiff</i> order conditions, and study their strong stability preserving property by writing them in Shu-Osher form. Temporal convergence of the proposed methods is theoretically analyzed under appropriate regularity assumptions. We further construct a class of fully discrete SRK and strong stability preserving SRK schemes with spectral accuracy in space by discretizing the expectation operator using the Sinc quadrature rule. Sufficient conditions for a SRK scheme to preserve the maximum principle are also briefly discussed. Ample numerical experiments are performed to demonstrate the accuracy and efficiency of the SRK schemes.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Probabilistic Representation for a Class of Exponential Elliptic Operators and Stochastic Runge-Kutta Methods for Semilinear Stiff Equations

  • 期刊论文

  • 发表日期：

    2026年2月21日

查看全文
  • Yabing Sun,
  • Quan Zhou

摘要

It is well known that exponential Runge-Kutta methods are widely used for solving semilinear stiff parabolic equations, where exponential elliptic operators are often encountered and the matrix exponential functions are inevitable. To develop a probabilistic approach for solving semilinear stiff parabolic equations without using the matrix exponential functions, in this paper, we give a probabilistic representation for a class of exponential elliptic operators under the periodic boundary condition via an expectation operator w.r.t. a diffusion process. Such a representation is a natural result of the nonlinear Feynman-Kac formula, from which we can write the solutions of a class of semilinear parabolic equations in forms of integral equation characterized by the expectation operator. Based on this integral equation, we propose a class of explicit time semidiscrete stochastic Runge-Kutta (SRK) methods for a class of semilinear parabolic equations, derive their stiff order conditions, and study their strong stability preserving property by writing them in Shu-Osher form. Temporal convergence of the proposed methods is theoretically analyzed under appropriate regularity assumptions. We further construct a class of fully discrete SRK and strong stability preserving SRK schemes with spectral accuracy in space by discretizing the expectation operator using the Sinc quadrature rule. Sufficient conditions for a SRK scheme to preserve the maximum principle are also briefly discussed. Ample numerical experiments are performed to demonstrate the accuracy and efficiency of the SRK schemes.