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Formula for 2nd order upwind scheme for non-uniform grids? |
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August 30, 2012, 04:57 |
Formula for 2nd order upwind scheme for non-uniform grids?
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#1 |
Senior Member
TWB
Join Date: Mar 2009
Posts: 414
Rep Power: 19 |
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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August 30, 2012, 05:57 |
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#2 |
Senior Member
Join Date: Dec 2011
Location: Madrid, Spain
Posts: 134
Rep Power: 15 |
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. Cheers, Michujo. |
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August 30, 2012, 18:25 |
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#3 | |
Senior Member
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,882
Rep Power: 73 |
Quote:
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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