[Jaydi_21]

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?

[Eifoehn4]

Dear Jaydi_21,

do you consider the "well balanced" property in your implementation?

Regards

[Jaydi_21]

Originally Posted by Eifoehn4

Dear Jaydi_21,

do you consider the "well balanced" property in your implementation?

Regards

I understand that this scheme is well-balanced because the discretization of the source term balances the fluxes. Otherwise, how can I consider that property? Is there any specific part of the paper that ensures the "well balanced" property?

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,

[Eifoehn4]

Originally Posted by Jaydi_21

Otherwise, how can I consider that property? Is there any specific part of the paper that ensures the "well balanced" property?

The whole paper deals with it, by using special Riemann solvers.

"The definition of the new numerical fluxes that include explicitly this new wave, leads as a consequence to the desired well-balance 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

[FMDenaro]

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 first-order accurate.



Therefore note that the scheme must be linear and also the time integration must be first order accurate.

[Jaydi_21]

Originally Posted by Eifoehn4

The whole paper deals with it, by using special Riemann solvers.

"The definition of the new numerical fluxes that include explicitly this new wave, leads as a consequence to the desired well-balance 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

Yes, you are right, the scheme is well balanced not only by upwinding the bottom topography source term, but also by adding an extra stationary wave. So, I don't understand what you meant by "Did you consider the well-balanced property in your implementation?".

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

[Jaydi_21]

Originally Posted by FMDenaro

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 first-order accurate.



Therefore note that the scheme must be linear and also the time integration must be first order accurate.

Theres is no additional checks? Just that is linear and first order time integration?

Thanks

[FMDenaro]

Originally Posted by Jaydi_21

Theres is no additional checks? Just that is linear and first order time integration?

Thanks

The key is that the theorem allows for non-linear 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.

[Eifoehn4]

Originally Posted by Jaydi_21

Yes, you are right, the scheme is well balanced not only by upwinding the bottom topography source term, but also by adding an extra stationary wave. So, I don't understand what you meant by "Did you consider the well-balanced property in your implementation?".

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

I did not read both papers in details, so I would suggest the following:

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

[Jaydi_21]

Originally Posted by Eifoehn4

I did not read both papers in details, so I would suggest the following:

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

Actually I was able to reproduce some of the examples in the paper, even including the non conservative term. The only difference with this case, is the wind (which is not included in the original paper). Do you think that could be the reason of the oscillations?

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

[piu58]

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.

[Eifoehn4]

Originally Posted by Jaydi_21

Actually I was able to reproduce some of the examples in the paper, even including the non conservative term. The only difference with this case, is the wind (which is not included in the original paper). Do you think that could be the reason of the oscillations?

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

If you have oscillations with first order, you won't fix them with second order. As i already told you, i think you have problems with inconsistent approximation of fluxes and sources.

Path-conservativ methods might help.

Regards

[Jaydi_21]

Originally Posted by Eifoehn4

If you have oscillations with first order, you won't fix them with second order. As i already told you, i think you have problems with inconsistent approximation of fluxes and sources.

Path-conservativ methods might help.

Regards

I found that if the mesh is aligned with the flow, I'm using a Cartesian mesh, those oscillations basically disappear. When the flow is 45 degrees oriented with respect to the mesh, I get the oscillations. Is there any way to improve that, other than orienting the mesh in the same direction than the flow? Can that be improved with a second order accurate version?

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

Thanks

[Jaydi_21]

Originally Posted by piu58

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.

For the problem I'm dealing with, I know the analytical solution and the oscillations are not part of it.


Thanks