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Usually you speak of a $n$'th order (accurate) method if your Taylor truncation error is of order $n+1$. This means your approximation is accurate up to order $n$ terms, and your errors are of order $n+1$. However, in FVM methods you often have no easy way of obtaining the truncation error of your formulation, since you reconstruct the numerical fluxes $F$ ...

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As far we are in an Incompressible Navier Stokes formulation, The velocity field and the pressure field are related in the domain and on the boundary (I suppose you are using the incompressible solver and hence a projection method algorithm to decouple the velocity and the pressure, in OpenFOAM should be PISO one). Taking the explicit velocity corrector for ...

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There is a lot of one can mention here, so I am going to kick off things. CFL is never a sufficient criterion for stability, but a necessary one. That said, just because you adhere the CFL condition, there is no guarantee that you get stable results. This makes absolute sense, since your methods can be arbitrarily bad. Looks to me you are trying not to ...

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To address the second question on the instability of the central scheme, it is helpful to consider the scalar advection equation $$u_t + c u_x = 0$$ which is a simplification of the convection equation $$u_t + \nabla \cdot (\boldsymbol{c} u) = 0$$ to one dimension and constant velocity $c \neq c(x)$. One can show that the central difference scheme  u_i^{(...

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Just happened to have a look at this old post and happened to notice that the given calculation in the accepted answer is scientifically completely (!) wrong such that I had to outline it in this post: The given computational burden is wrong by four orders of magnitude: A calculation which actually takes less than 2 hours and a half on a modern GPU would ...

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Probably a reasonably easy way to do it is to use this FCT code developed at US NRL, https://github.com/scivision/LCPFCT. It is designed for CFD problems of the kind that you are interested in, and it has some examples included. This example https://github.com/scivision/LCPFCT/blob/main/runfast2d.py looks rather similar to your problem, I would suggest ...

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