Numerical Integration of Stochastic Differential Equations

Systematic work on numerical solution of stochastic differential equations ( S D E S ) seems not to have kept pace with the considerable analytical developments. This parallels the lag which existed between the analytical and numerical study of ordinary differential equations near the turn of the century (which is perhaps understandable in view of the difficulty of implementing even straightforward algorithms at the time). In the last few years, there has been a burst of activity in performing Brownian dynamics computer simulations' to gain insight into motions in complex physical systems. Little attention seems to have been paid, though, to the systematic development of the numerical techniques in most of these works. In the present paper, the Runge-Kutta (RK) approximation for deterministic differential equations ( D D E S ) is extended to S D E S . Although we have not as yet explicitly considered other popular numerical schemes, we feel that the techniques utilized here should have wider applicability. For the sake of simplicity, several further restrictions are placed on the discussions in this paper. These, we believe, can ultimately be removed by fairly simple means. (i) We shall work only with a single equation rather than a set of n equations. It has been explicitly verified that the second-order approximation carries over in a straightforward manner to sets (and in our studies of polymers2 we used it for 600 simultaneous equations). 2289