Quadratic Invariant-Preserving Runge-Kutta Methods for the Numerical Solution of Stochastic Differential Equations
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In this paper we consider stochastic Runge-Kutta methods and expand some results from the deterministic theory of invariants to a stochastic setting. Specifically we find the necessary conditions for conservation of quadratic invariants, both for stohastic Runge-Kutta methods and stochastic, partitioned Runge-Kutta methods. Based on a theory of rooted trees and B-series we provide order conditions for stochastic, partitioned Runge-Kutta methods. We also construct and test some new methods.