An Introduction to Monte Carlo Methods in Statistical Physics.

Monte Carlo Solution of Boundary
Value Problems
Often have a differential equation with a boundary condition
∇
2
2
2
u = ∂ x / ∂ x + ∂ x / ∂ y = 0
2
2
Laplace' s equation
where u(r) = f on the boundary. Re-express as a finite
difference equation using a small increment ∆,
∇
2
u =
=
⇒ u
[ u ( x + ∆ , y ) + u ( x − ∆ , y ) + u ( x , y + ∆ ) + u ( x , y − ∆ ) − 4 u ( x , y ) ]
0
( x , y ) =
[ u ( x + ∆ , y ) + u ( x − ∆ , y ) + u ( x , y + ∆ ) + u ( x , y − ∆ ) ]/
4
/ ∆
2