Modeling Continuous-Time Random Processes in Digital Computer Simulations of Physical Systems

This dissertation addresses the problem of determining the correct relationship between the statistics of a continuous random process and the statistics of a continuous random process and the statistics of a discrete random process and the statistics of a discrete random process used to simulate the continuous random process. The findings of this research are directly applicable to the general by ordinary differential equations. It is shown that to ensure a faithful digital simulation of a continuous random process, the noise statistics of the random number generator must be set to values drastically different from the noise statistics of the continuous random process. Further, it is established that the relationship between the continuous and discrete statistics is a function of the integration method used in the digital simulation. The proper functional relationship between the discrete and continuous noise statistics is derived for the class of Runge-Kutta integrators, the 4th order Adams-Bashforth integrator, and the Adams-Moulton corrector formula. The derived relationships are applied to a specific problem and are demonstrated by stimulation. The stimulation results are compared to exact solutions. Additionally, the requirement for proper operation of a variable-step-size algorithm is developed.