Diagonally drift-implicit Runge-Kutta methods of weak order one and two for It\^o SDEs and stability analysis

TL;DR

This paper develops diagonally drift-implicit stochastic Runge-Kutta methods of weak order one and two for Itô SDEs, analyzing their stability properties and stability domains to guide step size selection.

Contribution

It introduces new diagonally drift-implicit stochastic Runge-Kutta methods of weak order one and two and analyzes their stability regions for better numerical approximation of SDEs.

Findings

Derived stability functions for the methods.

Compared stability domains with test equations.

Presented stability regions for various coefficients.

Abstract

The class of stochastic Runge-Kutta methods for stochastic differential equations due to R\"o{\ss}ler is considered. Coefficient families of diagonally drift-implicit stochastic Runge-Kutta (DDISRK) methods of weak order one and two are calculated. Their asymptotic stability as well as mean-square stability (MS-stability) properties are studied for a linear stochastic test equation with multiplicative noise. The stability functions for the DDISRK methods are determined and their domains of stability are compared to the corresponding domain of stability of the considered test equation. Stability regions are presented for various coefficients of the families of DDISRK methods in order to determine step size restrictions such that the numerical approximation reproduces the characteristics of the solution process.