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Unformatted text preview: NUMERICAL METHODS FOR SECOND-ORDER STOCHASTIC EQUATIONS KEVIN BURRAGE * , IAN LENANE * AND GRANT LYTHE Abstract. We seek numerical methods for second-order stochastic differential equations that accurately reproduce the stationary distribution for all values of damping. A complete analysis is possible for linear second-order equations (damped harmonic oscillators with noise), where the statis- tics are Gaussian and can be calculated exactly in the continuous-time and discrete-time cases. A matrix equation is given for the stationary variances and correlation for methods using one Gaussian random variable per timestep. The only Runge-Kutta 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. 60-08, 65c30. 1. Introduction. Newtons law states that acceleration is proportional to force. Consequently, second-order 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 second-order ordinary differential equations is one of the most mature branches of numerical analysis . The most ex- citing advances in recent decades have been the development of symplectic methods, capable of exactly preserving an energy-like quantity over very long times . In the stochastic setting, the long-time dynamics of a typical physical system is governed by fluctuation-dissipation, 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 . Numerical methods replace continuous-time with discrete-time 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 steady-state density as close as possible to that of the continuous-time system. The differential equations describing second-order systems contain a parameter known as damping. The steady-state 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