Chebyshev Methods with Discrete Noise: The $τ$-ROCK Methods (Journal of Computational Mathematics, 2018)
Stabilized or Chebyshev explicit methods have been widely used in the past to solve stiff ordinary differential equations. Making use of special properties of Chebyshev-like polynomials, these methods have favorable stability properties compared to standard explicit methods while remaining explicit. A new class of such methods, called ROCK, introduced in [Numer. Math., 90, 1-18, 2001] has recently been extended to stiff stochastic differential equations under the name S-ROCK [C. R. Acad. Sci. Paris, 345(10), 2007 and Commun. Math. Sci, 6(4), 2008]. In this paper we discuss the extension of the S-ROCK methods to systems with discrete noise and propose a new class of methods for such problems, the $τ$-ROCK methods. One motivation for such methods is the simulation of multi-scale or stiff chemical kinetic systems and such systems are the focus of this paper, but our new methods could potentially be interesting for other stiff systems with discrete noise. Two versions of the $τ$-ROCK methods are discussed and their stability behavior is analyzed on a test problem. Compared to the $τ$-leaping method, a significant speed-up can be achieved for some stiff kinetic systems. The behavior of the proposed methods are tested on several numerical experiments.
Keywords: Stiff stochastic differential equations; Runge-Kutta Chebyshev methods; Chemical reaction systems; tau-leaping method.