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 $\approx 1$
