So I inherited from some people a code that solves the advection-diffusion-reaction equation for a particular system. The original code was first implemented in 1D which worked fine in cartesian coordinates. However, now that I'm trying to implement it in 2d I'm having a few problems.
Separating the spatial discretization from the temporal through a method of lines and using a cartesian coordinates is relatively straightfrward to extend a model from 1D -> 2D because the spatial components are additive. However, the problem is that the genuine solutions to the equations I'm solving tends be kinda spherically symmetric, but the grid is cartesian so the solution tends to show grid orientation errors in the cartesian grid, i.e. the solution looks kinda "square" instead of looking like a "circle". A CPU costly solution to this grid orientation error is to make the grid finer, but it's getting to the point the computations become very slow.
I was able to solve the 2D grid orientation errors for the diffusion part of the equations by discretizing the laplacian into a 9-point stencil instead of the original 5-point stencil, the latter 5-point stencil is acquired by just naively adding the 1D laplacian discretations of the x and y components. The 9 point stencil adds spatially "diagonal" components to the discretization. However, the advection part still has some pretty bad grid orientation problems.
I was wondering if there is a "9-point stencil" analog to the 2D advection problem. Maybe a stencil that takes into account the "diagonal" cross-term spatial components somehow.
