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September 30, 2018, 14:26 
Are all first order upwind methods monotone?

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I'm using a Godunov type solver to solve the shallow water equations. This solver is very similar to the HLLC one. However when plotting the solution for a wind driven test case, I get some spurious oscillations. I've read that first order upwind methods are monotone, so they don't produce these spurious oscillations. How can I check if the solver I'm using is monotone and if it is, why am I getting these oscillations?
If you want to take a look at the solver, you can check it here: https://www.sciencedirect.com/scienc...21999112003464 If the solver is not monotone, can I get rid of these oscillations by making the solver second order accurate and using one of those limiters (TVD)? Or will I still have the oscillations even if I make it second order accurate? 

September 30, 2018, 14:41 

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Dear Jaydi_21,
do you consider the "well balanced" property in your implementation? Regards 

September 30, 2018, 15:32 
What property is that in the paper?

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I'm able to reproduce some other examples listed in the paper, which makes me think that my implementation is correct, but the wind driven test case, which is not listed in the paper, I'm not able to reproduce it. This is the wind driven test case I'm talking about: https://www.tandfonline.com/doi/abs/...21689209498945 While I'm able to get the correct pattern for the velocity field, I can see some spurious oscillations. Best, 

September 30, 2018, 16:38 

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"The definition of the new numerical fluxes that include explicitly this new wave, leads as a consequence to the desired wellbalance property. Therefore, the method is not constructed departing, as usual, from numerical relationships that force equilibrium among fluxes and source terms in cases of quiescent water." I think the question you should ask is: What specific equilibrium among fluxes and source terms is necessary to get the right solution in case of wind driven test? Regards 

September 30, 2018, 16:58 

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Filippo Maria Denaro
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The Godunov theorem states:
Linear numerical schemes for solving partial differential equations (PDE's), having the property of not generating new extrema (monotone scheme), can be at most firstorder accurate. Therefore note that the scheme must be linear and also the time integration must be first order accurate. 

September 30, 2018, 17:54 

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In the case of wind driven test, I'm including this term as part of the source terms, so I didn't change the solver. Should I do any specific consideration for the wind term? Or as you said, what specific equilibrium is required between the wind term and the fluxes? Thanks 

September 30, 2018, 17:55 

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Thanks 

September 30, 2018, 18:04 

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Filippo Maria Denaro
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The key is that the theorem allows for nonlinear scheme to be "monotone" even if they are of higher order. This the basis for the development for hystorical TVD schemes. I suggest the book of Leveque as reference. 

October 1, 2018, 02:17 

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1. Try to valide the conservativ part of the code. A conservativ method with first order in space and time should generally not generate oscillations. The same should hold for high order TVD schemes. 2. The paper also deals with species or concentration equations. It is generally known, that a first order method can generate oscillations if the fluxes in all equations in the system are not consistent. Perhaps some more "congrete" informations would be helpful. Regards 

October 2, 2018, 12:17 
That is my concern because the solver is well balanced!

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Might it be because the flow is not aligned with the grid? It's oriented 45 degrees with respect to the mesh. Finally, if I have those oscillations with the first order scheme, is it possible to correct the oscillations by making a second order accurate version of the solver? Thanks 

October 2, 2018, 15:10 

#11 
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Uwe Pilz
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Have you ever thought that at a wind driven problem oscillations may be part of the real solution? If yes, there are lots of reasons why these oscillations are numerical inforced.
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Uwe Pilz  Die der Hauptbewegung überlagerte Schwankungsbewegung ist in ihren Einzelheiten so hoffnungslos kompliziert, daß ihre theoretische Berechnung aussichtslos erscheint. (Hermann Schlichting, 1950) 

October 3, 2018, 17:17 

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Pathconservativ methods might help. Regards 

October 3, 2018, 18:19 

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I'm constructing the 2D solution from considering Riemann Problems in each separate direction, if that is of relevance! Thanks 

October 3, 2018, 18:21 

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Thanks 

Tags 
finite volume method, hllc, shallow water equations, source terms, upwind schemes 
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