Numerical Analysis of Systems of Ordinary and Stochastic Differential Equations

Artemiev, S. S.; Аверина, Т. А.

doi:10.1515/9783110944662

Cited by 69 publications

(54 citation statements)

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“…The standard concept of A-stability for numerical integrators of SDEs with additive noise is based on the test equation (4.16) (see, e.g., [2,26,37,40]). For a uniform time partition (t) h with t n = nh, n ∈ Z, let…”

Section: A-stabilitymentioning

confidence: 99%

“…Indeed, they involve the numerical solution of a system of nonlinear algebraic equations at each integration step. This algorithmic drawback is lessened in predictor-corrector methods (e.g., those based on explicit Runge-Kutta or Theta methods [52]) and linear-implicit methods (e.g., Rosenbrock methods [2]). However, the former are less stable than fully implicit and linear-implicit methods, while the latter ones still involve much more computational effort than explicit integrators, specially for high dimensional SDEs.…”

Section: Introductionmentioning

confidence: 99%

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High order local linearization methods: An approach for constructing A-stable explicit schemes for stochastic differential equations with additive noise

et al. 2010

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An approach for the construction of A-stable high order explicit strong schemes for stochastic differential equations (SDEs) with additive noise is proposed. We prove that such schemes also have the dynamical property that we call Random A-stability (RA-stability), which ensures that, for linear equations with stationary solutions, the numerical scheme has a random attractor that converges to the exact one as the step size decreases. Basically, the proposed integrators are obtained by splitting, at each time step, the solution of the original equation into two parts: the solution of a linear ordinary differential equation plus the solution of an auxiliary SDE. The first one is solved by the Local Linearization scheme in such a way that A-stability is guaranteed, while the second one is approximated by any extant scheme, preferably an explicit one that yields high order of convergence with low computational cost. Numerical integrators constructed in this way are called High Order Local LinearizaCommunicated by Anders Szepessy. 510 H. De la Cruz Cancino et al. tion (HOLL) methods. Various efficient HOLL schemes are elaborated in detail, and their performance is illustrated through computer simulations. Furthermore, meansquare convergence of the introduced methods is studied.

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Section: A-stabilitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

High order local linearization methods: An approach for constructing A-stable explicit schemes for stochastic differential equations with additive noise

et al. 2010

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“…Numerical methods for stochastic differential equations (SDEs) have been developed for several decades [8,18,19,28,36,38], and much progress in this subject has been recently reported [2,4,5,9,11,12,21,34]. In the progress, we are especially concerned with numerical methods with good stability properties.…”

Section: Introductionmentioning

confidence: 99%

Weak first- or second-order implicit Runge–Kutta methods for stochastic differential equations with a scalar Wiener process

Komori

2008

Journal of Computational and Applied Mathematics

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New fully implicit stochastic Runge-Kutta schemes of weak order 1 or 2 are proposed for stochastic differential equations with sufficiently smooth drift and diffusion coefficients and a scalar Wiener process, which are derivative-free and which are A-stable in mean square for a linear test equation in some general settings. They are sought in a transparent way and their convergence order and stability properties are confirmed in numerical experiments.

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“…[2,11,13,17,18,20,22,25,30,33,34,36] and the long-term behaviour of stochastic algorithms has been investigated as well, see e.g. [12,35,38] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Stationary distributions of Euler–Maruyama-type stochastic difference equations with Markovian switching and their convergence

Chen

Mao

2005

Journal of Difference Equations and Applications

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Stochastic differential equations with Markovian switching (SDEwMSs), one of the important classes of hybrid systems, have been used to model many physical systems that are subject to frequent unpredictable structural changes. The research in this area has been both theoretical and applied. Although the numerical methods for stochastic differential equations (SDEs) have been well studied, there are only a few results on the numerical solutions for SDEwMSs. By this manuscript we continue our series of papers on numerical solutions of SDEwMSs. The main aim of this paper is to investigate the stationary distributions of Euler-Maruyamatype stochastic difference equations with Markovian switching (EMSDEwMSs) and discuss their convergence.

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Numerical Analysis of Systems of Ordinary and Stochastic Differential Equations

Cited by 69 publications

References 0 publications

High order local linearization methods: An approach for constructing A-stable explicit schemes for stochastic differential equations with additive noise

High order local linearization methods: An approach for constructing A-stable explicit schemes for stochastic differential equations with additive noise

Weak first- or second-order implicit Runge–Kutta methods for stochastic differential equations with a scalar Wiener process

Stationary distributions of Euler–Maruyama-type stochastic difference equations with Markovian switching and their convergence

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