Static heat distribution problem

General problem:

$$ \dfrac{\partial}{\partial x}\left(k(x, z)\dfrac{\partial u}{\partial x}\right) + \dfrac{\partial}{\partial z}\left(k(x, z)\dfrac{\partial u}{\partial z}\right) = -f(x, z) $$

Simplified problem:

$$ \dfrac{\partial^2 u}{\partial x^2} + \dfrac{\partial^2 u}{\partial z^2} = - \dfrac{f(x, z)}{k} $$

Finite differences methods:

• Seidel
• Thomas (tridiagonal) - faster and more accurate

Monte-carlo methods:

• Fixed random walk - works fine
• Float random walk - fast but innacurate near heating source
• Semi-float random walk - works fine with step size $\approx 1$