Stochastic Runge-Kutta Methods for Preserving Maximum Bound Principle of Semilinear Parabolic Equations. Part II: Sinc Quadrature Rule (CSIAM Transactions on Applied Mathematics, 2025)
The maximum bound principle (MBP) is an important property for a largeclass of semilinear parabolic equations. To propose MBP-preserving schemes withhigh spatial accuracy, in the first part of this series, we developed a class of timesemidiscrete stochastic Runge-Kutta (SRK) methods for semilinear parabolic equations, and constructed the first- and second-order fully discrete MBP-preserving SRKschemes. In this paper, to develop higher order fully discrete MBP-preserving SRKschemes with spectral accuracy in space, we use the Sinc quadrature rule to approximate the conditional expectations in the time semidiscrete SRK methods and proposea class of fully discrete MBP-preserving SRK schemes with up to fourth-order accuracy in time for semilinear equations. Based on the property of the Sinc quadraturerule, we theoretically prove that the proposed fully discrete SRK schemes preserve theMBP and can achieve an exponential order convergence rate in space. In addition, wereveal that the conditional expectation with respect to the Bronwian motion in the timesemidiscrete SRK method is essentially equivalent to the exponential Laplacian operator under the periodic boundary condition. Ample numerical experiments are alsoperformed to demonstrate our theoretical results and to show the exponential orderconvergence rate in space of the proposed schemes.