Formula for 2nd order upwind scheme for non-uniform grids?
I've seen the formula for 2nd order upwind scheme for uniform grids as:
phi_e = 1.5*phi_P - 0.5*phi_W for u_e > 0
phi_e = 1.5*phi_E - 0.5*phi_EE for u_e < 0
Is there a formula meant for non-uniform grids as well?
Hi, you can try to derive it. Just take the Taylor expansions around the nodes E and EE, taking into account the different grid spacing for node i.
The formulas you showed is a particular case where the grid spacing is constant. Under this assumption, the grid spacings appearing in numerator and denominator of your expressions when you solve for phi_e cancel out and disappears from the equation. However, the different will appear in the expression of phi_e if you consider grid non-uniformity.
I hope it helps.
I suggest to construct a second degree polynomial on non-uniform stencil in such a way that if u_e>0 then Phi_W,Phi_P, Phi_E are involved (the counterpart for u_e<0 involves Phi_P,Phi_E, Phi_EE).
This way the use of Phi_P ensures better stability properties
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