Runge–Kutta Methods for the Strong Approximation of Solutions of Stochastic Differential Equations

Rößler, Andreas

doi:10.1137/09076636x

Cited by 172 publications

(152 citation statements)

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“…While by including further components of the Itô-Taylor expansion, higher order numerical schemes can be constructed, calculations of multiple stochastic integrals and their joint laws in the appeared higher order terms are very difficult and time-consuming. Therefore, a great deal of attention has been paid for developing derivative-free techniques such as SRKs due to their ease of programming, large stability regions, and flexible time-stepping strategy (Rößler 2010), see Appendix A.…”

Section: Methodsmentioning

confidence: 99%

“…where i = (Burrage and Burrage 1996), Rößler (2010) recently introduced a derivative-free SRK for path-wise approximation (strong order of 1.5) with the corresponding Butcher tableau for s = 4 shown as follows (SDRK4). This class of strong order 1.5 for the stochastic elements results in a strong order of 2 for the drift term.…”

Section: Appendix B: Stochastic Runge-kutta Schemementioning

confidence: 99%

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Turbulence-particle interactions under surface gravity waves

Paskyabi

2016

Ocean Dynamics

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The dispersion and transport of single inertial particles through an oscillatory turbulent aquatic environment are examined numerically by a Lagrangian particle tracking model using a series of idealised test cases. The turbulent mixing is incorporated into the Lagrangian model by the means of a stochastic scheme in which the inhomogeneous turbulent quantities are governed by a onedimensional k-ε turbulence closure scheme. This vertical mixing model is further modified to include the effects of surface gravity waves including Coriolis-Stokes forcing, wave breaking, and Langmuir circulations. To simplify the complex interactions between the deterministic and the stochastic phases of flow, we assume a time-invariant turbulent flow field and exclude the hydrodynamic biases due to the effects of ambient mean current. The numerical results show that the inertial particles acquire perturbed oscillations traced out as time-varying sinking/rising orbits in the vicinity of the sea surface under linear and cnoidal waves and acquire a non-looping single arc superimposed with the high-frequency fluctuations beneath the nonlinear solitary waves. Furthermore, we briefly summarise some recipes through the course of this paper on the implementation of the stochastic particle tracking models to realistically describe the drift and suspension of inertial particles throughout the water column.

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Section: Methodsmentioning

confidence: 99%

Section: Appendix B: Stochastic Runge-kutta Schemementioning

confidence: 99%

Turbulence-particle interactions under surface gravity waves

Paskyabi

2016

Ocean Dynamics

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“…Now, by applying the colored rooted tree theory for Itô SDEs given in [4,13,14], order conditions for the coefficients of the SRK method (2.3) can be easily calculated, making use of the vector e = (1, . .…”

Section: A Stochastic Runge-kutta Methods For Sdesmentioning

confidence: 99%

“…The SRK method (2.3) is covered by the general class of SRK methods proposed in [13]. Therefore, Proposition 5.2 in [13] based on the colored rooted tree analysis can be applied, which directly results in the order conditions given in Theorem 2.1. 2 …”

Section: A Stochastic Runge-kutta Methods For Sdesmentioning

confidence: 99%

A Runge-Kutta method for index 1 stochastic differential-algebraic equations with scalar noise

2011

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The paper deals with the numerical treatment of stochastic differential-algebraic equations of index one with a scalar driving Wiener process. Therefore, a particularly customized stochastic Runge-Kutta method is introduced. Order conditions for convergence with order 1.0 in the mean-square sense are calculated and coefficients for some schemes are presented. The proposed schemes are stiffly accurate and applicable to nonlinear stochastic differential-algebraic equations. As an advantage they do not require the calculation of any pseudo-inverses or projectors. Further, the mean-square stability of the proposed schemes is analyzed and simulation results are presented bringing out their good performance.

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“…Both strong and weak stochastic Runge-Kutta methods (SRK) for diffusion SDE have been considered in detail in [8,9,16,25,27], and the order condition has been obtained by colored trees analysis. In this paper, we derive a two-stage explicit SRK of strong order 1 for jump-diffusion SDE.…”

Section: Introductionmentioning

confidence: 99%