The Stochastic$\alpha$ Method: A Numerical Method for Simulation of Noisy Second Order Dynamical Systems
Nagalinga Rajan and Soumyendu Raha

Source CMES: Computer Modeling in Engineering & Sciences, Vol. 23, No. 2, pp. 91-116, 2008
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Keywords Second order Stochastic Differential Equations, Multiplicative Noise, Multiple It\^o Integrals, Numerical Dissipation
Abstract The article describes a numerical method for time domain integration of noisy dynamical systems originating from engineering applications. The models are second order stochastic differential equations (SDE). The stochastic process forcing the dynamics is treated mainly as multiplicative noise involving a Wiener Process in the It\^o sense. The developed numerical integration method is a drift implicit strong order 2.0 method. The method has user-selectable numerical dissipation properties that can be useful in dealing with both multiplicative noise and stiffness in a computationally efficient way. A generalized analysis of the method including the multiplicative noise is presented. Strong order convergence, user-selectable numerical dissipation and stability properties are established in the analysis of the method. The concept of stochastic contractivity has been developed in this context. The integration method is illustrated with numerical examples of noisy mechanical systems. The method addresses the need for higher strong order convergent stochastic schemes for efficient simulation and design analysis of stiff and highly oscillatory engineering systems with multiplicative noise.
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