I'm also wondering, in the case of the IBVP for the laplace equation with nonconstant coefficients..how does that affect the random walk scheme? Sure, we should still be able to use a monte carlo method for such a game, but the probabilities of going in each direction change depending on the coefficient...Actually. He does give an example in the book I linked, but sadly never said exactly how to determine such values of $/frac{1}{2(1+/sin(x_{j}))$. In fact, the author failed to give any other explanation of finite difference methods with equations with nonconstant coefficients

Thank you for the enormous amount of info you've given me, it's going to be a huge amount of help since there is extremely little literature on the application of games and solutions to PDE's. From what I've been reading about the 1-D heat equation is that the Crank-Nicolson is essentially the best way of approximating it, in most of the books that I've found at my library they've introduced the finite difference method for it, and then basically introduce this new method that does not have stability problems. Unless I'm not understanding something, why wouldn't I just employ the other method?