ABSTRACT
An underlying concept of the probabilistic or Monte Carlo solution of differential equations is the random walk. Different types of random walk lead to different Monte Carlo methods (MCMs). This chapter discusses the fixed random walk. An interesting illustration of random walks in one dimension is a random walk along Madison Avenue. The fixed random walk MCM can be used to solve Poisson's equation. The chapter extends the random walk Monte Carlo procedures to potential problems with axisymmetric geometries. It is also important to consider an extension of the MCM for the solution of problems with discrete homogeneities, that is, homogeneous media separated by interfaces. One major disadvantage of the fixed random MCM is that it computes the potential at only one point at a time. Another major drawback of the fixed random walk is that it is slow because a large number of steps per walk are required to reach the boundary due to the small, fixed step size.
