Solving the Nonlinear Algebraic Equations with Monte Carlo Method

Abstract

Many problems of rarefied gas dynamics can be solved by imitation of the collision molecules process [see, for example, Bird (1994), and Illner and Neunzert (1987)]. It is known that this process is described with a good precision by the equation with quadratic nonlinearity (the Boltzman equation). The relation between the Boltzman equation and branching and collision processes was studied in the book [Ermakov, Nekrutkin and Sipin (1989)] and a number of following papers [for example, Goliandina (1996), and Nekrutkin and Tur (1997)]. It worth to note that a great number of papers were devoted to this relation. However, the problem of the relation between general type equations with quadratic nonlinearity and random processes was studied relatively small. Especially, numerical techniques for the related processes simulation were not enough investigated. At the same time many physical processes are described by equations with quadratic, or, in general, polynomial nonlinearity. The important example is the Navier-Stokes equations.

It is known also that many simulation techniques for the solution of equations are based on the Neumann-Ulam scheme (N. U. scheme). And it is of a great interest to study the relation of this scheme with known simulation techniques for the solution of problems of rarefied gas dynamic.

This work continues the research of the certain kind of Monte Carlo algorithms for solving the equations with polynomial nonlinearity. The authors have limited themselves with a detailed considering of the systems of algebraic quadratic equations.