Stochastic Runge–Kutta methods for multi-dimensional Itô stochastic differential algebraic equations

Abstract

In this paper, we discuss the numerical solutions to index 1 stochastic differential algebraic equations. We introduce a new class of weak second-order stochastic Runge–Kutta methods for finding the numerical approximate solutions to multi-dimensional stochastic differential algebraic equations. A four-stage stiffly accurate stochastic Runge–Kutta methods for approximating analytical solutions to index 1 stochastic differential algebraic equations are derived. By colored rooted tree analysis, the order conditions for the stochastic Runge–Kutta methods of order two satisfying the weak convergence is obtained. The scalar test equations are considered to obtain the mean-square stability and the T-stability of weak second-order stochastic Runge–Kutta methods. Finally, some numerical illustrations are provided to prove the theoretical findings.

Keywords

• Stochastic differential algebraic equations
• Stochastic Runge–Kutta methods
• T-stability
• Mean-square stability
• Stiffly accurate
• Weak convergence