Numerical Methods for Second-order Stochastic Equations
نویسندگان
چکیده
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 statistics 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.
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