I know that the criterion of stability for an explicit solution to the heat equation:(adsbygoogle = window.adsbygoogle || []).push({});

[itex]\frac{\partial T}{\partial t}=D\frac{\partial^2 T}{\partial x^2}[/itex]

is

[itex]\Delta t <\frac{1}{2}\frac{\Delta x^2}{D}[/itex]

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

[itex]\frac{\partial T}{\partial t}=D\frac{\partial^2}{\partial x^2}\left(\frac{T}{P(D)}\right)[/itex]

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:

[itex]\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})[/itex]

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.

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# Stability criteria of heat-like equation

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