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The Stochastic α Method: A Numerical Method for Simulation of Noisy Second Order Dynamical Systems

Nagalinga Rajan, Soumyendu Raha1
Supercomputer Educationand Research Centre, Indian, Institute of Science, Bangalore 560012, India. Email:, The second author is the corresponding author.

Computer Modeling in Engineering & Sciences 2008, 23(2), 91-116.


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 Itô 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.


Second order Stochastic Differential Equations, Multiplicative Noise, Multiple Itô Integrals, Numerical Dissipation

Cite This Article

Rajan,, N. (2008). The Stochastic α Method: A Numerical Method for Simulation of Noisy Second Order Dynamical Systems. CMES-Computer Modeling in Engineering & Sciences, 23(2), 91–116.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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