Random Choice and Related Methods

Abstract

In 1965, Glimm [141] introduced the Random Choice Method (RCM) as part of a constructive proof of existence of solutions to a class of non—linear systems of hyperbolic conservation laws. In 1976, Chorin [73] successfully implemented a modified version of the method, as a computational technique, to solve the Euler equations of Gas Dynamics. In essence, to implement the RCM one requires (i) exact solutions of local Riemann problems and (ii) a random sampling procedure to pick up states to be assigned to the next time level. As we shall see, there is a great deal of commonality between the RCM and the Godunov method presented in Chap. 6. Both schemes use the exact solution of the Riemann problem, although Godunov’s method can also be implemented using approximate Riemann solvers, as we shall see in Chaps. 9 to 12. The two methods differ in the way the local Riemann problem solutions are utilised to march to the next time level: the Godunov method takes an integral average of local solutions of Riemann problems, while the RCM picks a single state, contained in the local solutions, at random. The random sampling procedure is carried out by employing a sequence of random numbers. The statistical properties of these random numbers have a significant effect on the accuracy of the Random Choice Method.