# BTCS-like method for heat conduction in unstructured triangular grid

I want to write a simple simulation for heat conduction in a unstructured triangular mesh.

I already made it work for a structured rectangular grid with the ADI method, but now I need more complex geometries.

With $d_x=\frac{\kappa \Delta t}{(\Delta x)^2}$ and $d_y=\frac{\kappa \Delta t}{(\Delta y)^2}$ you can say:

$$d_x T_{i+1,j}^{n+1}+d_x T_{i-1,j}^{n+1}-(2d_x+2d_y+1)T_{i,j}^{n+1}+d_y T_{i,j-1}^{n+1}+d_y T_{i,j+1}^{n+1}=-T_{i,j}^n\ ,$$

and you basically have a penta-diagonal equations system.

For the triangular grid, the only thing I came up with till now is to calculate the heat-flux-density over every edge, sum them up for every cell and add them to the temperature of the cell. For me this seems to be a explicit method, which (if it follows the same behaviour than the FTCS approximation) should be pretty instable.

Thats why I want to come up with an implicit method, but till now, I'm not able to.

Has anybody some advice or some estimation on the stability of my explained method?

• Hi Hendrik, welcome to this forum. Unfortunately you cannot do that. Finite differences are based on Taylor expansions, and thus, they need an orthogonal grid. On the other hand, there are other "finite difference methods" that are based on functions that "turns" the non-orthogonal grid to an orthogonal one. They are finite elements, and they are really suitable for example for elliptic equations (heat equation) what you need. – HBR Aug 2 '17 at 14:21
• You can define Finite Differences on unstructured meshes. See this reference, for example. Taken from this post. – nicoguaro Aug 2 '17 at 19:39
• Solving the heat equation by computing heat fluxes along the edges in a (Delaunay) triangular mesh is very much akin to a Finite Volume scheme on the Voronoi dual of the triangular mesh, using the grid nodes as the Voronoi centers. If you construct a Voronoi diagram of the mesh, then you can get the area of each face between adjacent cells. Combined with an appropriate definition of the heat flux, you can then use this to define a consistent finite volume method for your problem. – Tyler Olsen Aug 2 '17 at 20:34
• I will note that for my approach, you probably have to be very careful to get 2nd order accuracy, which is desirable considering how easy it is to attain at least that level of accuracy using finite element methods for this problem. – Tyler Olsen Aug 2 '17 at 20:36