I want to solve the following set of coupled advection-diffusion equations: $$ \frac{\partial f}{\partial t}=\nabla\cdot(\kappa\nabla f)+\nabla\cdot(\boldsymbol{u}f)+s_f(g), $$ $$ \frac{\partial g}{\partial t}=\nabla\cdot(\kappa\nabla g)+\nabla\cdot(\boldsymbol{u}g)+s_g(f), $$ in polar coordinates $(r,\theta)$. Using finite elements is natural in that case. However, to help for stability and computation time, I would like to have a grid equally spaced in $\cos\theta$ and the zeros of Chebyshev polynomials for $r$. I never tried using finite volume methods so I am a bit unsure of what I am doing. My question is this: what is the best way to go about this? Can I simply use Chebyshev interpolation and derivation for values on volume faces? Or can I solve the system for the Chebyshev coefficients instead? I plan to use a fully explicit time stepping scheme (probably Runge-Kutta second order)
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