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Unformatted text preview: NUMERICAL METHODS FOR SECONDORDER STOCHASTIC EQUATIONS KEVIN BURRAGE * , IAN LENANE * AND GRANT LYTHE Abstract. We seek numerical methods for secondorder stochastic differential equations that accurately reproduce the stationary distribution for all values of damping. A complete analysis is possible for linear secondorder equations (damped harmonic oscillators with noise), where the statis tics are Gaussian and can be calculated exactly in the continuoustime and discretetime cases. A matrix equation is given for the stationary variances and correlation for methods using one Gaussian random variable per timestep. The only RungeKutta method with a nonsingular tableau matrix in the class that gives the exact steady state density for all values of damping is the implicit midpoint rule. Numerical experiments comparing the implicit midpoint rule with Heun and leapfrog methods suggest that the qualitative behavior is similar to the linear case for nonlinear equations with additive or multiplicative noise. Key words. damped harmonic oscillators with noise, stationary distribution, stochastic Runge Kutta methods, implicit midpoint rule, multiplicative noise. AMS subject classifications. 6008, 65c30. 1. Introduction. Newtons law states that acceleration is proportional to force. Consequently, secondorder differential equations are common in scientific applica tions, in the guise of Langevin, Monte Carlo, Molecular or Dissipative parti cle dynamics [1, 2], and the study of methods for secondorder ordinary differential equations is one of the most mature branches of numerical analysis [3]. The most ex citing advances in recent decades have been the development of symplectic methods, capable of exactly preserving an energylike quantity over very long times [4]. In the stochastic setting, the longtime dynamics of a typical physical system is governed by fluctuationdissipation, so that the amount of time spent in different regions of phase space is controlled by the stationary density. The stationary density can have a relatively simple explicit expression even when the dynamics is highly nonlinear [5]. Numerical methods replace continuoustime with discretetime dynamics, generating values at times t ,t 1 ,... . Usually t n +1 t n is a fixed number t . The criterion for a good numerical method that will be examined in this work is that its discrete time dynamics has a steadystate density as close as possible to that of the continuoustime system. The differential equations describing secondorder systems contain a parameter known as damping. The steadystate density is independent of damping, but time dependent quantities and the usefulness or otherwise of numerical algorithms are strongly dependent. As the damping tends to infinity the system becomes first order....
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 Fall '06
 gallivan

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