- #1
Hypatio
- 147
- 1
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.
\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.