DG methods with linear time-dependent Advection-Diffusion PDE. Positive initial values and negative projection onto FE space and negative solution.

[aydar...@gmail.com]

Dear all,

I am having troubles with obtaining negative values both for projection of initial values and obtaining the solution in 2D.
I use symmetric interior penalty method with upwind stabilization for advection term with first order DG elements.

When I use VectorTools::project(...) the (positive) initial value function onto my solution vector, I obtain some negative values.
I tried both very smooth initial conditions (gaussian) and very sharp (circle with some positive value inside and 0 everywhere else in the domain).
The negative values are near sharp edges.

Then, I used VectorTools::interpolate(...), which did give me non-negative projection of the initial condition.
But the scheme still produces negative values for solution. As the scheme marches in time forward, the negative values reduce in absolute value
(both with positive or negative initial values).
Furthermore, the negative values do not seem to bother time marching. After many time-steps (depending on penalty parameter, smoothness and "positivity" of the initial value), negative values become very small in absolute value

This reduction happens for both strictly parabolic case and strictly hyperbolic case, as well as a mix of both.
Plotting the solution over time even with negative values does not "look" wrong, as one does not even notice that negative values are there, except if having a colorbar.

I am not sure where things went wrong. 

Thanks for any help.

Best,

Aydar

[Wolfgang Bangerth]

> When I use VectorTools::project(...) the (positive) initial value function
> onto my solution vector, I obtain some negative values.

You need to expect this if the function you are projecting is sufficiently
non-smooth (on length scales of the mesh size). The projection onto a
(continuous) finite element space is really not very different to a truncated
Fourier series: you get Gibbs phenomenon with over and undershoots. There is
nothing you can do if you insist on (linear) projections.

> Then, I used VectorTools::interpolate(...), which did give me non-negative
> projection of the initial condition.
> But the scheme still produces negative values for solution.

Most methods actually do. It is very difficult to construct non-negative (or
monotonous) numerical schemes for the advection equation. That said, even
schemes that are not monotonous typically converge -- i.e., they converge to
the correct solution, but the finite dimensional approximations just don't
satisfy one of the physical constraints you have (that the solution is
non-negative).

Best
W.

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Wolfgang Bangerth email: bang...@math.tamu.edu
www: http://www.math.tamu.edu/~bangerth/

[aydar...@gmail.com]

Most methods actually do. It is very difficult to construct non-negative (or
monotonous) numerical schemes for the advection equation. That said, even
schemes that are not monotonous typically converge -- i.e., they converge to
the correct solution, but the finite dimensional approximations just don't
satisfy one of the physical constraints you have (that the solution is
non-negative).

I use upwind method for linear advection. I know that Godunov numerical flux (or upwind for linear advection) is monotone, at least in the context of Finite Volume Methods.
This should imply that linear upwind discretization with DG method should as well be monotonous, should it not?

Can you give me a short suggestion about how one can get monotonicity?
How much effect does adding mass (replacing negative values by 0) have on scheme?

Thank you.
Best,
A

 

[Wolfgang Bangerth]

On 11/09/2015 11:46 AM, aydar...@gmail.com wrote:
> I use upwind method for linear advection. I know that Godunov numerical flux
> (or upwind for linear advection) is monotone, at least in the context of
> Finite Volume Methods.
> This should imply that linear upwind discretization with DG method should as
> well be monotonous, should it not?

I have no idea -- I don't know enough about this field. Do you do mass lumping?

> Can you give me a short suggestion about how one can get monotonicity?

I don't know how one can do this. People have spent their whole lives
developing schemes that are monotonous. They do exist for low order methods
but I don't know the appropriate entry points into the literature.

[Sungho Yoon]

It's advisable to use the lumped mass matrix if you would project the initial solution on the mesh or use the smoothed functional as bengerth  told , otherwise the negative values are obtained on projection.

Sungho

2015년 11월 9일 월요일 오전 9시 57분 14초 UTC-8, bangerth 님의 말:

[Praveen C]

Hello Aydar

This can be said to be a lack of stability in maximum norm or violation of maximum principle. There are some methods available to improve this in DG schemes, see the works of Zhang and Shu. You can construct schemes so that cell average values satisfy maximum principle and this is enough to ensure stable and accurate solutions even for non-linear problems. I do not know of any methods available to ensure that the full polynomial solution satisfies maximum principle. For affine solutions on simplex elements this is doable since if you control the vertex values, then the entire solution will be bounded.

For linear problems, small violations may not be an issue, But in some other problems, like compressible flows, density and pressure have to strictly positive, otherwise the computations will break down.

This is still an unsolved problem but there is slow progress, atleast for scalar problems, see e.g., this recent paper for a different approach

https://www.researchgate.net/publication/270764210_On_discrete_maximum_principles_for_discontinuous_Galerkin_methods

Best

praveen

[Aydar Uatay]

Thank you for reply. I will try more to figure out this problem.  

                                www: http://www.math.tamu.edu/~bangerth/

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[Aydar Uatay]

Thank you for reply. Mass lumping seems to work. 

Best,

Aydar

[Aydar Uatay]

Thank you for replies. Mass lumping seems to work for initial projection. 
I will be trying to work out Praveen's suggestion. 

Thank you for help.

Best,

Aydar 

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