# Stability criteria of heat-like equation

1. Nov 9, 2014

### Hypatio

I know that the criterion of stability for an explicit solution to the heat equation:

$\frac{\partial T}{\partial t}=D\frac{\partial^2 T}{\partial x^2}$
is
$\Delta t <\frac{1}{2}\frac{\Delta x^2}{D}$

however, what is the stability criterion for an equation of the form

$\frac{\partial T}{\partial t}=D\frac{\partial^2}{\partial x^2}\left(\frac{T}{P(D)}\right)$

where P(D) indicates that P depends on the value of D, which varies arbitrarily in space.

I would like to solve this equation with explicit finite differences, so I will have a term of the form:

$\frac{\partial T}{\partial t}=\frac{D+D_L}{2 \Delta x^2}(\frac{C_L}{P_L}-\frac{C}{P})+\frac{D+D_R}{2\Delta x^2}(\frac{C_R}{P_R}-\frac{C}{P})$

where subscripts L and R indicate relative position (left and right of the point of calculation).

How can I figure out the maximum timestep allowed.

2. Nov 14, 2014