Changing the domain of a 3D Finite Difference code from cube to sphere

Question

I have an explicit FD (Finite Difference) code for diffusion/heat on a PDE in a cuboid domain, and it works fine. I would like to update the discretized equations and change the code so as to solve the problem on a sphere. From some preliminary searches, I can't seem to find any examples of FD on spheres using rectangular coordinates.

I have seen some in spherical coordinates, but that may require too many changes to the code (if this is the only way, please advice on how best to proceed)

$$\frac{\partial U}{\partial t} = \alpha \nabla^2U\quad in\enspace\mathbb{R}^3$$ $$U(x,y,z,0)=T$$ where T is a constant, and is also the boundary condition applied on all faces of the cube / the surface area of the sphere.

Question: Is it possible to adapt my code for a cube, to work on a sphere, and remain in rectangular coordinates? If so, how do I approach this?

Edit: (more context) This is a homogenous heat equation, where the boundary condition is a constant temperature (say T=100 degrees) applied on the entire surface area of the sphere (for the cube it was also applied on all 6 faces). The initial condition is the same heat source on the surface, while the entire insides of the sphere was initially at zero.

Answer 1

That is certainly possible. You can reformulate your problem to use $\Delta + V$ instead of $\Delta$ as an operator, where $V=V(x,y,z)=V(\vec{r})$ is a potential defined in a way suitable to tackle your problem. For instance, if I were looking for eigenmodes of a disc or a sphere by solving the Helmholtz equation using finite differences I'd define the potential as

$V(\vec{r}) = \begin{cases}0 & \vec{r}< r_0\\ V_0 & else\end{cases}$

with $V_0\rightarrow\infty$ being essentially a (very) large number. This should, in the case of a Helmholtz problem, force the resulting solution to be ~0 outside of the sphere with radius $r_0$. For a heat equation you can do something similar (depending on the boundary condition).