Forward Euler method for solving PDE’s in CompuCell3D.¶
We present more complete derivations of explicit finite difference scheme for diffusion solver in “Introduction to Hexagonal Lattices in CompuCell3D” (http://www.compucell3d.org/BinDoc/cc3d_binaries/Manuals/HexagonalLattice.pdf).
In CompuCell3D most of the solvers uses explicit schemes (Forward Euler method) to obtain PDE solutions. Thus for the diffusion equation we have:
In a discretetized form we may write:
where to save space we used shorthand notation:
and similarly for other coordinates.
After rearranging terms we get the following expression:
where the sum over index \(i\) goes over neighbors of point \((x,y,z)\) and the neighbors will have the following concentrations: \(c(x+\delta x, t)\), \(c(y+\delta y, t)\), \(c(z+\delta z, t)\) .