Non-standard finite-differences schemes for reaction–diffusion equations in curvilinear coordinates

In this work, we extend our previous results (Alvarez-Ramirez, J., Valdes-Parada, F., Alvarez, J., & Ochoa-Tapia, A. (2007). A Green’s function formulation for finite-differences schemes. Chemical Engineering Science, 62, 3083–3091) to the case of reaction–diffusion equations in cylindrical and spherical coordinates. Contrary to standard finite-differences (FD) schemes, the Green’s function formulation incorporates domain curvature effects by weighting the central differences with non-linear metrics, indeed, logarithmic distance for cylindrical and inverse distance for spherical coordinates. It is shown that the Green’s function formulation can be reduced to standard FD schemes for small values of the grid spacing. Numerical simulations show that the modifications introduced improve the performance of traditional FD schemes, specially for coarse grids.