Study on random walk and its application to solution of heat conduction equation by Monte Carlo method

In this paper optimized and fast numerical method based on Monte Carlo simulation are introduced to achieve several algorithms for solving the steady state and transient heat conduction problems with any prescribed convection boundary conditions and interior thermal source distribution. The methods are mesh-less and thus can solve problems with complex geometries, non-uniform heat source or porous bodies with no inherent limitation. In the proposed approach the temperature of any point in the studying body can be found independently from other points without solution of related system of linear algebraic equations. As part of this study, an effort was undertaken to summarize various random walk methods in Monte Carlo. The Monte Carlo methods that are introduced in this paper use Semi Floating Random Walk (SFRW) and Full Floating Random Walk (FFRW) to estimate temperature in the domain of the definition. Also, Fixed Random Walk (FRW) method is applied to solve transient heat conduction equation. Each method has its own unique features that will be discussed during this work. It was observed that FFRW method is the fastest method to steady state conditions and also FRW method has capability to run both steady state and transient heat conduction equation. It was showed that the results are in good agreement with those of the finite difference method.