The split-step backward Euler method for linear stochastic delay differential equations

AbstractIn this paper, the numerical approximation of solutions of linear stochastic delay differential equations (SDDEs) in the Itô sense is considered. We construct split-step backward Euler (SSBE) method for solving linear SDDEs and develop the fundamental numerical analysis concerning its strong convergence and mean-square stability. It is proved that the SSBE method is convergent with strong order γ=12 in the mean-square sense. The conditions under which the SSBE method is mean-square stable (MS-stable) and general mean-square stable (GMS-stable) are obtained. Some illustrative numerical examples are presented to demonstrate the order of strong convergence and the mean-square stability of the SSBE method

Errors of linear multistep methods and Runge-Kutta methods for singular perturbation problems with delays

AbstractThis paper is concerned with the error analysis of linear multistep methods and Runge-Kutta methods applied to some classes of one-parameter stiff singularly perturbed problems with delays. We derive the global error estimates of A(α)-stable linear multistep methods and algebraically and diagonally stable Runge-Kutta methods with Lagrange interpolation procedure. Numerical experiments confirm our theoretical analysis