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Formula for 2nd order upwind scheme for non-uniform grids?

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Old   August 30, 2012, 03:57
Default Formula for 2nd order upwind scheme for non-uniform grids?
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Hi,

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?

Thanks
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Old   August 30, 2012, 04:57
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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 \Delta x_i 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 \Delta x disappears from the equation. However, the different \Delta x_i will appear in the expression of phi_e if you consider grid non-uniformity.

I hope it helps.

Cheers,
Michujo.
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Old   August 30, 2012, 17:25
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Quote:
Originally Posted by quarkz View Post
Hi,

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?

Thanks
Be careful that upwind schemes are usually first, third, fifth ... order accurate... the reason is to have a local truncation error that has a dissipative behaviour ... the above formulas are based on a linear extrapolation of the value, that is highly unstable in general.

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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