Implicit stochastic Runge-Kutta methods for stochastic differential equations. (English) Zbl 1048.65005
Summary: We construct implicit stochastic Runge-Kutta (SRK) methods for solving stochastic differential equations of Stratonovich type. Instead of using the increment of a Wiener process, modified random variables are used. We give convergence conditions of the SRK methods with these modified random variables. In particular, the truncated random variable is used. We present a two-stage stiffly accurate diagonal implicit SRK (SADISRK2) method with strong order 1.0 which has better numerical behaviour than extant methods. We also construct a five-stage diagonal implicit SRK method and a six-stage stiffly accurate diagonal implicit SRK method with strong order 1.5. The mean-square and asymptotic stability properties of the trapezoidal method and the SADISRK2 method are analysed and compared with an explicit method and a semi-implicit method. Numerical results are reported for confirming convergence properties and for comparing the numerical behaviour of these methods.
MSC:
| 65C30 | Numerical solutions to stochastic differential and integral equations |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
| 60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 34F05 | Ordinary differential equations and systems with randomness |
