Appropriate monotone flux for WENO3

selig5576 · November 7, 2017, 11:58

I've recently successfully implemented WENO-3 for the advection term in my incompressible solver and am interested in implementing a monotone flux scheme, such as Godunov. For the Godunov it adjusts the stencil via

$F_{i+1/2,j}=$ $\min_{u_{L} \le u \le u_{R}} F(u),u_{L} \le u_{R}$
$\max_{u_{R} \le u \le u_{L}} F(u) ,u_{L} > u_{R}$

I have naively implemented this with my WENO scheme and it seems to work, however I have read that calculating the Godunov flux is difficult as it reduces to solving an optemization problem for each quadrature. But since I am on a structured grid, do I need to worry about solving an optemization problem?

praveen · November 7, 2017, 23:11

It is not a grid related issue. If you have to find the min and max as written in your formula, that can be expensive in general.

If F(u) is convex, then you can write down the solution explicitly. In other situations, you may be able to find the min/max explicitly, in which it is ok. But if you have to apply some numerical optimization technique to find the min/max, then it can become expensive.

Also, the formula you write for Godunov flux is only for a scalar conservation law.

selig5576 · November 8, 2017, 09:02

Praveen, thank you for the reply. The Godunov method is indeed slowing my code down. In the case of using the Local Lax Friedrichs (or Rusanov scheme), given I am in a incompressible regime, how would one calculate $\alpha$ . In compressible flows it is calculated as $\alpha = \max\left({|u_{i,j}| + c_{i,j}, |u_{i,j}| - c_{i,j}}\right)$ , i.e. the Eigenvalues.

EDIT: For the burgers equation \alpha is calculated as $\alpha = \max(u_{i}, u_{i+1})$ , i.e. differentiating the flux. Applying the same procedure for the F and G flux for the u-momentum I would have $\alpha = \max(u_{i,j}, u_{i+1,j})$ for F flux and $\alpha = \max(u_{i,j},u_{i,j+1},v_{i,j},v_{i,j+1})$ for the G flux. Given there are u and v velocity components I would imagine it would have to be modified for the G flux so it could accommodate the relevant ranges.

The definition of the Local Lax Friedrichs I am following is the one in Shu's monograph:

$f(u_{i+1/2,j}^{+},u_{i+1,j}^{-}) = \frac{1}{2} \left(F(u_{i+1/2,j}^{+}) + F(u_{i+1/2,j}^{-}) - \alpha (u_{i+1/2,j}^{+} - u_{i+1/2,j}^{-})\right)$

arungovindneelan · November 10, 2017, 07:15

Quote:

Originally Posted by selig5576

In the case of using the Local Lax Friedrichs (or Rusanov scheme), given I am in a incompressible regime, how would one calculate $\alpha$ . In compressible flows it is calculated as $\alpha = \max\left({|u_{i,j}| + c_{i,j}, |u_{i,j}| - c_{i,j}}\right)$ , i.e. the Eigenvalues.

Sorry, I don't have experience on WENO on incompressible flows. In incompressible flow also u can calculate the speed of sound, may end up in slightly wrong result if u assume density is constant.

I'm just curious to know: whether these kinds of flux calculating algorithm is required for incompressible flow? can't we use physical flux directly? I have studied on papers that this kind of density based algorithms are very slow in the incompressible region but can be overcome by pre-conditioner.

The computational cost of WENO is much higher than simple interpolation. WENO is a good interpolation scheme where we have shard corners but those structures may not possible in incompressible flow. Please answer whether it is worth or not. Is there any special incompressible case it is required?

selig5576 · November 10, 2017, 11:41

I've only just implemented the WENO-3 scheme, however I do believe its worth while as a scheme for the advection term.

I have not noticed any slow down using the LLxF scheme for the advection term. I am using LLxF so I could accommodate the different wind directions without using an upwind type method. I am still in the testing phase, however I can send you my code if you are interested. I've tested my solver on the lid driven cavity test and am getting pretty good results. I have also red in literature being have used MUSCL for the advection term, which would probably be much faster than WENO or ENO.

November 7, 2017, 11:58	Appropriate monotone flux for WENO3	#1
selig5576 Senior Member Selig Join Date: Jul 2016 Posts: 213 Rep Power: 10	I've recently successfully implemented WENO-3 for the advection term in my incompressible solver and am interested in implementing a monotone flux scheme, such as Godunov. For the Godunov it adjusts the stencil via $F_{i+1/2,j}=$ $\min_{u_{L} \le u \le u_{R}} F(u),u_{L} \le u_{R}$ $\max_{u_{R} \le u \le u_{L}} F(u) ,u_{L} > u_{R}$ I have naively implemented this with my WENO scheme and it seems to work, however I have read that calculating the Godunov flux is difficult as it reduces to solving an optemization problem for each quadrature. But since I am on a structured grid, do I need to worry about solving an optemization problem?

November 7, 2017, 23:11		#2
praveen Super Moderator Praveen. C Join Date: Mar 2009 Location: Bangalore Posts: 342 Blog Entries: 6 Rep Power: 18	It is not a grid related issue. If you have to find the min and max as written in your formula, that can be expensive in general. If F(u) is convex, then you can write down the solution explicitly. In other situations, you may be able to find the min/max explicitly, in which it is ok. But if you have to apply some numerical optimization technique to find the min/max, then it can become expensive. Also, the formula you write for Godunov flux is only for a scalar conservation law. selig5576 likes this. __________________ http://cpraveen.github.io http://twitter.com/cfdlab http://github.com/cpraveen

November 8, 2017, 09:02	Monotone flux	#3
selig5576 Senior Member Selig Join Date: Jul 2016 Posts: 213 Rep Power: 10	Praveen, thank you for the reply. The Godunov method is indeed slowing my code down. In the case of using the Local Lax Friedrichs (or Rusanov scheme), given I am in a incompressible regime, how would one calculate $\alpha$ . In compressible flows it is calculated as $\alpha = \max\left({\|u_{i,j}\| + c_{i,j}, \|u_{i,j}\| - c_{i,j}}\right)$ , i.e. the Eigenvalues. EDIT: For the burgers equation \alpha is calculated as $\alpha = \max(u_{i}, u_{i+1})$ , i.e. differentiating the flux. Applying the same procedure for the F and G flux for the u-momentum I would have $\alpha = \max(u_{i,j}, u_{i+1,j})$ for F flux and $\alpha = \max(u_{i,j},u_{i,j+1},v_{i,j},v_{i,j+1})$ for the G flux. Given there are u and v velocity components I would imagine it would have to be modified for the G flux so it could accommodate the relevant ranges. The definition of the Local Lax Friedrichs I am following is the one in Shu's monograph: $f(u_{i+1/2,j}^{+},u_{i+1,j}^{-}) = \frac{1}{2} \left(F(u_{i+1/2,j}^{+}) + F(u_{i+1/2,j}^{-}) - \alpha (u_{i+1/2,j}^{+} - u_{i+1/2,j}^{-})\right)$ Last edited by selig5576; November 8, 2017 at 10:03.

November 10, 2017, 11:41	Weno	#5
selig5576 Senior Member Selig Join Date: Jul 2016 Posts: 213 Rep Power: 10	I've only just implemented the WENO-3 scheme, however I do believe its worth while as a scheme for the advection term. I have not noticed any slow down using the LLxF scheme for the advection term. I am using LLxF so I could accommodate the different wind directions without using an upwind type method. I am still in the testing phase, however I can send you my code if you are interested. I've tested my solver on the lid driven cavity test and am getting pretty good results. I have also red in literature being have used MUSCL for the advection term, which would probably be much faster than WENO or ENO.

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