Maximum Principles for a Class of Quasilinear Parabolic Equations and Stochastic Runge-Kutta Schemes

Source abstract

In this paper, we develop a probabilistic approach to study a practically desired property for a class of quasilinear parabolic equations in their applications, namely a time-invariant maximum principle. Based on their probabilistic representations via coupled forward and backward stochastic differential equations (FBSDEs), we first theoretically derive the maximum principle for these equations by using the comparison theorem of coupled FBSDEs. Then, by writing the solutions of quasilinear parabolic equations in the form of stochastic integral equations without containing any differential operator, we employ a stochastic Runge-Kutta approach to develop a class of high-order accurate temporal discretization methods, which are characterized by the operator of conditional expectations with respect to a diffusion process. Under some reasonable constraints on the time step, we prove the preservation of maximum principle of the proposed methods and analyze their convergence in time. We also construct the first- and second-order fully discrete schemes by solving the diffusion process using explicit derivative-free Itô-Taylor schemes and discretizing the operator of conditional expectations with the Gaussian quadrature rule and the linear interpolation. By studying the discretized operator of conditional expectations, we obtain the maximum principle-preserving property of the fully discrete schemes and rigorously establish their error estimates. Various numerical experiments are performed to verify the theoretical results.