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I looked at a similar question to find out how to discretize the heat equation in 2 spacial dimensions at the boundary between two materials. I am required to use explicit method (forward-time-centered-space) to solve. I believe I have arrived at a reasonable approximation with the following:

\begin{align}\frac{T^{k+1}_{i,j}-T^k_{i,j}}{\Delta t}&=\frac1{h^2}\left[\left(\alpha^k_{i+\frac12,j}-\alpha^k_{i-\frac12,j}\right)\left(T^k_{i+\frac12,j}-T^k_{i-\frac12,j}\right)+\left(\alpha^k_{i,j+\frac12}-\alpha^k_{i,j-\frac12}\right)\left(T^k_{i,j+\frac12}-T^k_{i,j-\frac12}\right)\right.\\ &\quad\left.+\alpha^k_{i,j}\left(T^k_{i-1,j}-2T^k_{i,j}+T^k_{i+1,j}\right)+\alpha^k_{i,j}\left(T^k_{i,j-1}-2T^k_{i,j}+T^k_{i,j+1}\right)\right] \end{align}

where any index $\pm\frac12$ represents the average of the value at current index and the one right in front of or behind, i.e.: $$\frac12(T_{i\pm1}+T_{i}).$$

$k$ is the time index, $i,j$ are the spacial indices, $h$ is a grid size, and $\Delta t$ is a time step.

This was determined through the generalized heat equation in $x$ and $y$ dimensions $$\frac{\partial T}{\partial t}=\nabla\cdot\left(\alpha\nabla T\right).$$

I have two questions:

  1. Is the discretized form of the heat equation above correct? If not, how do I fix it?
  2. Is it possible to use a nondimensional time step $\tau = \frac{\alpha\Delta t}{h^2}$ for this problem even though the $\alpha$ is changing across materials? If the nondimensional time step is changing, how would I then extract the actual solution in space and time after it is finished solving?
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  • $\begingroup$ For #2, I'd suspect that you'd use the maximum, as it's done in the CFL condition. $\endgroup$
    – Kyle Kanos
    Apr 13, 2018 at 1:22
  • $\begingroup$ Should I then use a constant time step $\tau$ ? And then scale it by whichever $\alpha$ range I'm in? $\endgroup$
    – AAC
    Apr 13, 2018 at 2:58
  • $\begingroup$ At the discontinuity between the materials, you have $$k_1\left(\frac{\partial T}{\partial x}\right)_1=k_2\left(\frac{\partial T}{\partial x}\right)_2$$where k is the thermal conductivity, and the subscripts refer to the two sides of the interface. $\endgroup$ Apr 13, 2018 at 14:56

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I discuss in that answer that the FDM for the 1D diffusion equation with variable coefficient ends up being, $$ \frac{T^{n+1}_i-T^n_i}{dt}=\frac{1}{dx^2}\left[\alpha_{i+\frac{1}{2}}\left(T^n_{i+1}-T^n_{i}\right)-\alpha_{i-\frac{1}{2}}\left(T^n_i-T^n_{i-1}\right)\right] $$ with $\alpha_{i+\frac12}=\left(\alpha_{i+1}+\alpha_{i}\right)/2$. Extending this 2D should be direct: just add $j$ subscript to all terms, then add a similar $\left[\cdots\right]/dy^2$ term. So it seems you've got a few extra terms, at least at an initial, cursory look. I also discuss the Crank-Nicolson version in this answer.

For the timestep, $dt$, you need to satisfy the Courant-Freidrich-Lewy condition (CFL) for diffusion equations: $$ CFL\leq\max\left(\alpha\right)\cdot\frac{dt}{\left(dx^2+dy^2\right)} $$ where, typically, $CFL=1/2$. So there will be one time step and its value depends on only the maximum value of the (variable) diffusion coefficient.

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