Construction and mean-square stability analysis of a new family of stochastic Runge-Kutta methods

This research paper investigates the convergence and stability of two diagonal drift-implicit second-order stochastic Runge-Kutta methods for weak approximation of systems containing three-dimensional drift and noise terms in Itô stochastic differential equations. The first method is based on the approach presented by Debrabant and Rößler (2008) [5], while the second method utilizes a Butcher table that, to the best of our knowledge, has not been used in previous research. We compare the convergence and stability of both methods and analyze their respective stability regions. The results show that the method using the newly introduced Butcher table is not only reliable but also highly efficient. •We propose a numerical method for SDEs that utilizes a new Butcher table.•We demonstrate the advantages of the new method over a more standard numerical approach.•We obtain stability regions for different values of the SDE parameters.•The new scheme achieves a substantial improvement in accuracy across various step sizes.