Stability analysis of high order Runge–Kutta methods for index 1 stochastic differential-algebraic equations with scalar noise

Abstract

In this paper, a new class of implicit stochastic Runge-Kutta (SRK) methods is constructed for numerically solving systems of index 1 stochastic differential-algebraic equations (SDAEs) with scalar multiplicative noise. By applying rooted tree theory analysis, the family of coefficients of the proposed methods of order 1.5 are calculated in the mean-square sense. In particular, we derive some four-stage stiffly accurate semi-implicit SRK methods for approximating index 1 SDAEs with scalar noise. For these methods, first, MS-stability functions, applied to a scalar linear test equation with multiplicative noise, are calculated. Then, their regions of MS-stability are compared with the corresponding MS-stability region of the original SDE. Accordingly, we illustrate that the proposed schemes seems to have good stability properties. Numerical results will be presented to check the convergence order and computational efficiency of the new methods.