@Article{CSIAM-AM-5-2,
author = {Yabing, Sun and Zhao, Weidong},
title = {Stochastic Runge-Kutta Methods for Preserving Maximum Bound Principle of Semilinear Parabolic Equations. Part I: Gaussian Quadrature Rule},
journal = {CSIAM Transactions on Applied Mathematics},
year = {2024},
volume = {5},
number = {2},
pages = {390--420},
abstract = {<p style="text-align: justify;">In this paper, we propose a class of stochastic Runge-Kutta (SRK) methods
for solving semilinear parabolic equations. By using the nonlinear Feynman-Kac formula, we first write the solution of the parabolic equation in the form of the backward
stochastic differential equation (BSDE) and then deduce an ordinary differential equation (ODE) containing the conditional expectations with respect to a diffusion process.
The time semidiscrete SRK methods are then developed based on the corresponding
ODE. Under some reasonable constraints on the time step, we theoretically prove the
maximum bound principle (MBP) of the proposed methods and obtain their error estimates. By combining the Gaussian quadrature rule for approximating the conditional expectations, we further propose the first- and second-order fully discrete SRK
schemes, which can be written in the matrix form. We also rigorously analyze the
MBP-preserving and error estimates of the fully discrete schemes. Some numerical experiments are carried out to verify our theoretical results and to show the efficiency
and stability of the proposed schemes.</p>},
issn = {2708-0579},
doi = {https://doi.org/10.4208/csiam-am.SO-2023-0032},
url = {https://global-sci.com/article/91023/stochastic-runge-kutta-methods-for-preserving-maximum-bound-principle-of-semilinear-parabolic-equations-part-i-gaussian-quadrature-rule}
}