Stabilized Numerical Methods for Stochastic Differential Equations driven by Diffusion and Jump-Diffusion Processes

Blumenthal, Adrian

doi:10.5075/epfl-thesis-6771

Cited by 1 publication

(1 citation statement)

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“…Well known explicit stabilized methods are the Runge-Kutta-Chebyshev (RKC) methods [41,45,47], the DUMKA methods [31,32,36] and the Runge-Kutta orthogonal Chebyshev (ROCK) [1,7] methods (note that the first-order RKC and ROCK scheme coincide). More recently, the first-order RKC (or ROCK) scheme has been extended to SDEs, yielding the S-ROCK family [3,4,6,10] and higher order extensions in [8]. For mean-square stable problems the S-ROCK scheme introduced in [6] represents an important improvement over the Euler-Maruyama method thanks to its improved stability properties (it does however not preserve the optimal stability domain of the first-order RKC method).…”

Section: Introductionmentioning

confidence: 99%

Optimal explicit stabilized postprocessed $τ$-leap method for the simulation of chemical kinetics

Abdulle,

Gander,

de Souza

2021

Preprint

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The simulation of chemical kinetics involving multiple scales constitutes a modeling challenge (from ordinary differential equations to Markov chain) and a computational challenge (multiple scales, large dynamical systems, time step restrictions). In this paper we propose a new discrete stochastic simulation algorithm: the postprocessed second kind stabilized orthogonal τ -leap Runge-Kutta method (PSK-τ -ROCK). In the context of chemical kinetics this method can be seen as a stabilization of Gillespie's explicit τ -leap combined with a postprocessor. The stabilized procedure allows to simulate problems with multiple scales (stiff), while the postprocessing procedure allows to approximate the invariant measure (e.g. mean and variance) of ergodic stochastic dynamical systems. We prove stability and accuracy of the PSK-τ -ROCK. Numerical experiments illustrate the high reliability and efficiency of the scheme when compared to other τ -leap methods.

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