divergence with higher order scheme
I am doing DNS and am in process of implementing higher order schemes. I am done with implementing it in momentum equation etc.
However, when it comes to calculate divergence in a cell, i.e. Div = du/dx+dv/dy+dw/dz, I am getting a bit confused whether I should implement higher order scheme here as well or I should keep it just second order accurate.
I am asking it because if 2nd order accurate,
du/dx = (u[i+1]-u[i])/(x[i+1]-x[i]) etc. on staggered grids. It gives a feeling of the rate of mass accumulation in the cell, a physically meaningful quantity.
However, if we change this to 4th order discretization,
du/dx = f(u[i+2], u[i+1], u[i], u[i-1]) etc. which doesn't look like giving mass accumulation in the cell in discrete manner.
Could anyone please point out which one should be preferred?
Usually you should keep the same order of discretization on all spatial derivatives if you want to have a pure 4 order scheme let's say. Otherwise your DNS will be only 2nd order.
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