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Ableidinger, M.; Buckwar, E.

Weak stochastic Runge-Kutta Munthe-Kaas methods for finite spin ensembles. (English) Zbl 1367.65012

Appl. Numer. Math. 118, 50-63 (2017).
Summary: In this article we construct weak Runge-Kutta Munthe-Kaas methods for a finite-dimensional version of the stochastic Landau-Lifshitz equation (LL-equation). We formulate a Lie group framework for the stochastic LL-equation and derive regularity conditions for the corresponding system of stochastic differential equations on the Lie algebra. Using this formulation we define weak Munthe-Kaas methods based on weak stochastic Runge-Kutta methods (SRK methods) and provide sufficient conditions such that the Munthe-Kaas methods inherit the convergence order of the underlying SRK method. The constructed methods are fully explicit and preserve the norm constraint of the LL-equation exactly. Numerical simulations are provided to illustrate the convergence order as well as the long time behaviour of the proposed methods.

Cited in 3 Documents

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
34F05 Ordinary differential equations and systems with randomness

Keywords:

stochastic Landau-Lifshitz-Gilbert equation; Lie group methods; Munthe-Kaas methods; weak stochastic Runge-Kutta methods; structure preservation; Stratonovich integral; numerical examples; convergence
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