Communications in Mathematical Sciences

S-ROCK methods for stiff Ito SDEs

A. Abdulle and T. Li

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In this paper, we present a class of explicit numerical methods for stiff Itô stochastic differential equations (SDEs). These methods are as simple to program and to use as the well-known Euler-Maruyama method, but much more efficient for stiff SDEs. For such problems, it is well known that standard explicit methods face step-size reduction. While semi-implicit methods can avoid these problems at the cost of solving (possibly large) nonlinear systems, we show that the step- size reduction phenomena can be reduced significantly for explicit methods by using stabilization techniques. Stabilized explicit numerical methods called S-ROCK (for stochastic orthogonal Runge- Kutta Chebyshev) have been introduced as an alternative to (semi-) implicit methods for the solution of stiff stochastic systems. In this paper we discuss a genuine Itô version of the S-ROCK methods which avoid the use of transformation formulas from Stratonovich to Itô calculus. This is important for many applications. We present two families of methods for one-dimensional and multi-dimensional Wiener processes. We show that for stiff problems, significant improvement over classical explicit methods can be obtained. Convergence and stability properties of the methods are discussed and numerical examples as well as applications to the simulation of stiff chemical Langevin equations are presented.

Article information

Commun. Math. Sci., Volume 6, Number 4 (2008), 845-868.

First available in Project Euclid: 18 December 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H35: Computational methods for stochastic equations [See also 65C30] 65C30: Stochastic differential and integral equations 65C20: Models, numerical methods [See also 68U20] 65L20: Stability and convergence of numerical methods

Stiff stochastic differential equations multiscale systems explicit stochastic methods Runge-Kutta Chebyshev methods stiff chemical Langevin equation


Abdulle, A.; Li, T. S-ROCK methods for stiff Ito SDEs. Commun. Math. Sci. 6 (2008), no. 4, 845--868.

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