Weak Second Order Conditions for Stochastic Runge--Kutta Methods

Tocino, Ángel; Vigo‐Aguiar, Jesús

doi:10.1137/s1064827501387814

Cited by 61 publications

(49 citation statements)

References 4 publications

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“…It can be studied by means of a modified action [6], reminiscent of modified equations used to understand finite-time convergence [7,8]. Such techniques provide a procedure for devising numerical methods, as well as analyzing them, by matching terms in powers of Δt [9,10,11]. Here, by imposing conditions obtained from analysis of linear equations, we construct explicit PRK methods that approximate the stationary density with high-order accuracy.…”

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confidence: 99%

Accurate Stationary Densities with Partitioned Numerical Methods for Stochastic Differential Equations

Burrage¹,

Lythe²

2009

SIAM J. Numer. Anal.

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Abstract. We devise explicit partitioned numerical methods for second-order-in-time scalar stochastic differential equations, using one Gaussian random variable per timestep. The construction proceeds by analysis of the stationary density in the case of constant-coefficient linear equations, imposing exact stationary statistics in the position variable and absence of correlation between position and velocity; the remaining error is in the velocity variable. A new two-stage "reverse leapfrog" method has good properties in the position variable and is symplectic in the limit of zero damping. Explicit new "Runge-Kutta leapfrog" methods are constructed, sharing the property that q n+1 = qn + 1 2 (pn + p n+1 )Δt, whose mean-square velocity order increases with the number of stages.

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confidence: 99%

Accurate Stationary Densities with Partitioned Numerical Methods for Stochastic Differential Equations

Burrage¹,

Lythe²

2009

SIAM J. Numer. Anal.

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“…Classical approaches for getting high weak order numerical schemes for stochastic differential equations are based on weak Taylor approximation or Runge-Kutta type methods [8,12]. For example, weak second order methods were proposed by Milstein [29,30], Platen [36], Mackevicius [27], Talay [43] (see also [22,32]) and Tocino and Vigo-Aguiar [45]. We mention also the extrapolation methods of Talay and Tubaro [44] and of [23] that combines methods with different stepsizes to achieve higher weak order convergence.…”

Section: W [M] (T))mentioning

confidence: 99%

High Weak Order Methods for Stochastic Differential Equations Based on Modified Equations

Abdulle¹,

Cohen²,

Vilmart³

et al. 2012

SIAM J. Sci. Comput.

128

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Abstract. Inspired by recent advances in the theory of modified differential equations, we propose a new methodology for constructing numerical integrators with high weak order for the time integration of stochastic differential equations. This approach is illustrated with the constructions of new methods of weak order two, in particular, semi-implicit integrators well suited for stiff (meansquare stable) stochastic problems, and implicit integrators that exactly conserve all quadratic first integrals of a stochastic dynamical system. Numerical examples confirm the theoretical results and show the versatility of our methodology.

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“…Various higher order weak methods have been considered in the literature [17,27]. For example, weak second order methods were proposed by Milstein [24,25], Platen [28], Talay [34] and Tocino and Vigo-Aguiar [37], and more recently Runge-Kutta type methods of Rößler [30]. We mention also the extrapolation methods of Talay and Tubaro [35] and of [18] that combines methods with different stepsizes to achieve higher weak order convergence.…”

Section: Weak Stochastic Integratorsmentioning

confidence: 99%

Weak Second Order Explicit Stabilized Methods for Stiff Stochastic Differential Equations

Abdulle¹,

Vilmart²,

Zygalakis³

2013

SIAM J. Sci. Comput.

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To cite this version:Assyr Abdulle, Gilles Vilmart, Konstantinos Zygalakis. Weak second order explicit stabilized methods for stiff stochastic differential equations. SIAM J. Sci. Comput., Sociey for Industrial and Applied Mathematics, 2013, 35 (4) We introduce a new family of explicit integrators for stiff Itô stochastic differential equations (SDEs) of weak order two. These numerical methods belong to the class of one-step stabilized methods with extended stability domains and do not suffer from the stepsize reduction faced by standard explicit methods. The family is based on the standard second order orthogonal Runge-Kutta Chebyshev methods (ROCK2) for deterministic problems. The convergence, mean-square and asymptotic stability properties of the methods are analyzed. Numerical experiments, including applications to nonlinear SDEs and parabolic stochastic partial differential equations are presented and confirm the theoretical results.

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Weak Second Order Conditions for Stochastic Runge--Kutta Methods

Cited by 61 publications

References 4 publications

Accurate Stationary Densities with Partitioned Numerical Methods for Stochastic Differential Equations

Accurate Stationary Densities with Partitioned Numerical Methods for Stochastic Differential Equations

High Weak Order Methods for Stochastic Differential Equations Based on Modified Equations

Weak Second Order Explicit Stabilized Methods for Stiff Stochastic Differential Equations

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