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I'm trying to solve the 2d heat equation in a radially symmetric domain, numerically using the Crank-Nicolson method. i.e.

[tex]\dfrac{\partial u}{\partial t} = D\left( \dfrac{\partial^2u}{\partial r^2}+\dfrac{1}{r}\dfrac{\partial u}{\partial r}\right)[/tex]

Applying the Crank-Nicolson method basically results in a recurrence relation:

[tex](-q+\frac{z}{r})u_{i-1,j+1} + (1+2q)u_{i,j+1} - (q+\frac{z}{r})u_{i+1,j+1} = (q-\frac{z}{r})u_{i-1,j} + (1-2q)u_{i,j} + (q+\frac{z}{r})u_{i+1,j} [/tex]

where [tex]q=\dfrac{\delta t}{2(\delta r)^2}[/tex], [tex]z=\dfrac{\delta t}{4\delta r}[/tex] and [tex]u_{i,j}[/tex] is the solution at [tex]r=i\delta r, t=j\delta t[/tex].

You can write this into a matrix equation of the form [tex]Au\{j+1\}=Bu\{j\}[/tex], which basically enables you to solve the problem. This all works very nicely for the 1d heat equation (the same differential equation but without the first spatial derivative). However, I now have the problem that the solution doesn't conserve mass. That is, given some initial condition across [tex]r\in (0,R)[/tex], the sum of the solution points decreases over time.

Clearly, if the column sums of A and B are the row vector of 1s, then I am guaranteed mass conservation. However, this is not the case for my Crank-Nicolson implementation here. This results from the [tex]\frac{z}{r}[/tex] changing values and signs between neighbouring rows, which means they don't cancel.

Anyone got any comments? Is there a way of solving these parabolic PDEs in radially symmetric domains that preserves the mass conservation law? Or is my implementation incorrect maybe?

Your assistance would be greatly appreciated,

Neil

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# Mass conservation in radially symmetric parabolic PDE problems

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