[alibaig1991]

Dear all,

Consider a 5x5 mesh of uniform spacing, with velocity U=[1,1]. What should be the restrictions on the weight values on the nodes for scheme to be an upwind scheme?

I have tried to develop one such scheme. The figure below shows x-, and y-coordinates, and the weight values at each node. (Absolute values of weight are shown in the figure.) Are the weight values correct for an upwind scheme?


Thank you.

[FMDenaro]

I don't understand your post, you should write the discretization you adopted to see what you are doing. Is the scheme 1d and applied separately on x and y direction? what the order of the accuracy?

[alibaig1991]

Originally Posted by FMDenaro

I don't understand your post, you should write the discretization you adopted to see what you are doing. Is the scheme 1d and applied separately on x and y direction? what the order of the accuracy?

Dear Mr. FMDenaro,

Thank you for replying.

Regarding you statement about whether it is 1D or 2D, the scheme is 2D and is not applied separately on x and y direction.

I am trying to create a new upwind scheme. The image attached in the first post shows the weight values at different points in a 2D stencil.

For 1D stencils, the weight of upstream point should be greater than the weight of downstream point, provided that they are at same distance from ref. point, for an scheme to be classified as an upwinded scheme. The scheme I have developed satisfies this condition.

My question is what are the conditions that a scheme should satisfy so that it can be classified as an upwind scheme in case of 2D.

[FMDenaro]

Originally Posted by alibaig1991

Dear Mr. FMDenaro,

Thank you for replying.

Regarding you statement about whether it is 1D or 2D, the scheme is 2D and is not applied separately on x and y direction.

I am trying to create a new upwind scheme. The image attached in the first post shows the weight values at different points in a 2D stencil.

For 1D stencils, the weight of upstream point should be greater than the weight of downstream point, provided that they are at same distance from ref. point, for an scheme to be classified as an upwinded scheme. The scheme I have developed satisfies this condition.

My question is what are the conditions that a scheme should satisfy so that it can be classified as an upwind scheme in case of 2D.

That does not depend on the weight of the nodes but on the fact that the stencil is upwinded in a 2D way, according to both sign of u and v components. A transport condition at unitary Courant number should be implied.
You should show the computational molecule around a point i,j.

[alibaig1991]

Dear Sir,

Originally Posted by FMDenaro

That does not depend on the weight of the nodes but on the fact that the stencil is upwinded in a 2D way, according to both sign of u and v components.

The reference point (point at which I want to approximate derivatives) is the the center point in the first figure, i.e., (0,0).

I guess what you mean is that I should include more points from upstream side in computational molecule than downstream side. This is not what I am trying to do. I am changing weights of points such that the weight of upstream points are greater in magnitude than the weights of downstream points. So, in this context, what should be the conditions that these weight values should satisfy?

Thank you.

[FMDenaro]

I assume you are discretizing the equation

df/dt +u df/dx + v df/dy = 0

Therefore, it seems you are using a central stencil for d/dx and d/dy and using non-symmetric weight on the nodes, right?

As I wrote, you need that your scheme is:
- consistent
- stable
- transportive properties (check for v=0 and cfl=1 that you get the exact solution, then invert the case)

However, I don't understand how you can get a real 2D molecule using FD. That would be possible if you adopt a FV formulation where the line integrals are involved, similarily to QUICK/QUICKEST schemes

[alibaig1991]

Originally Posted by FMDenaro

I assume you are discretizing the equation

df/dt +u df/dx + v df/dy = 0

Therefore, it seems you are using a central stencil for d/dx and d/dy and using non-symmetric weight on the nodes, right?

As I wrote, you need that your scheme is:
- consistent
- stable
- transportive properties (check for v=0 and cfl=1 that you get the exact solution, then invert the case)

However, I don't understand how you can get a real 2D molecule using FD. That would be possible if you adopt a FV formulation where the line integrals are involved, similarily to QUICK/QUICKEST schemes

Dear Sir,

At last we are on same page.

Yes, the weight values are non-symmetric and I am using a meshless method.

When using non-symmetric weights for 1D problem, the weights for upstream points should be greater than weights for downstream points. This is the condition that weight values must satisfy in order to achieve upwinding.

There must be similar conditions for non-symmetric 2D weights. What are those conditions? What should be the relation between weight values such that I can say that there is no cross-wind diffusion?

Thank you so much for your time.

[FMDenaro]

You should introduce terms like d2f/dxdy, d2f/dx2 and d2f/dy2 in a tensor anti-diffusion terms...That is somehow what happens in a Lax-Wendroff formulation.

However, I don't understand what you are doing as you wrote "meshless" method. Then, what about the time integration?

[alibaig1991]

Originally Posted by FMDenaro

You should introduce terms like d2f/dxdy, d2f/dx2 and d2f/dy2 in a tensor anti-diffusion terms...That is somehow what happens in a Lax-Wendroff formulation.

However, I don't understand what you are doing as you wrote "meshless" method. Then, what about the time integration?

Dear Sir,

The weights for discretization of spatial derivatives are obtained using the meshless method, whereas, fro time integration, I shall use Euler explicit method.

[FMDenaro]

Originally Posted by alibaig1991

Dear Sir,

The weights for discretization of spatial derivatives are obtained using the meshless method, whereas, fro time integration, I shall use Euler explicit method.

Using first order explicit method I doubt you can get a numerically stable solution if the downwind weights are of the same order of magnitude of the upwind ones. The hyperbolic character of the equations requires some constraint about the direction of the characteristic lines. Then, you definitely need to assess that the scheme is consistent.
Have you tried to do some numerical test?

[alibaig1991]

Originally Posted by FMDenaro

Using first order explicit method I doubt you can get a numerically stable solution if the downwind weights are of the same order of magnitude of the upwind ones. The hyperbolic character of the equations requires some constraint about the direction of the characteristic lines. Then, you definitely need to assess that the scheme is consistent.
Have you tried to do some numerical test?

Dear Sir,

The weights for upstream and downstream are not of the same order of magnitude, at least for 1D.

I have not tried any numerical test, however, I have solved a few numerical examples for 1D case with high Pe numbers, and the results are encouraging. For very high Pe number, the conventional meshless method results in spurious oscillations. The method I developed successfully suppressed those oscillations.

Thank you.

[FMDenaro]

Originally Posted by alibaig1991

Dear Sir,

The weights for upstream and downstream are not of the same order of magnitude, at least for 1D.

I have not tried any numerical test, however, I have solved a few numerical examples for 1D case with high Pe numbers, and the results are encouraging. For very high Pe number, the conventional meshless method results in spurious oscillations. The method I developed successfully suppressed those oscillations.

Thank you.

Sorry but this is not clear, you are solving a convection-diffusion problem or only a convection problem?
If you are solving a convection-diffusion problem the weights you showed consider both effects. That implies the discretization of the laplacian that is not upwinded.

[alibaig1991]

Originally Posted by FMDenaro

Sorry but this is not clear, you are solving a convection-diffusion problem or only a convection problem?
If you are solving a convection-diffusion problem the weights you showed consider both effects. That implies the discretization of the laplacian that is not upwinded.

Dear Sir,

I am solving highly convective convection-diffusion equation in 2 dimensions. The image in first post shows the weight values for first derivative only (d/dx). These weight values are upwinded (are they? This is the main question).

For second derivatives, I will use symmetric weights, which are not difficult to obtain.

[FMDenaro]

Originally Posted by alibaig1991

Dear Sir,

I am solving highly convective convection-diffusion equation in 2 dimensions. The image in first post shows the weight values for first derivative only (d/dx). These weight values are upwinded (are they? This is the main question).

For second derivatives, I will use symmetric weights, which are not difficult to obtain.

To be honest, I don't think you are following any rigorous methodology in your approach, at least reading your posts. I don't know if I can be of help this way.
Now you wrote your computational stencil is only for the df/dx. But, according to the convection equation this partial derivative can be upwinded only along the x-direction following the path (x-u*dt). Therefore, there is no theoretical reason to produce a 2D stencil for this term.
The 2D stencil would appear if you consider the vector term v.Grad f and you consider the directional derivative df/ds along the vector direction (x -v*dt).

A further approach where a 2D stencil appear is if you use the integral form of the equation and you reconstruct the numerical flux to be integrated along the FV face.

You can see some upwinded schemes in Eq.s(25) in:
https://www.researchgate.net/publica...mes_simulation

[alibaig1991]

Originally Posted by FMDenaro

To be honest, I don't think you are following any rigorous methodology in your approach, at least reading your posts. I don't know if I can be of help this way.
Now you wrote your computational stencil is only for the df/dx. But, according to the convection equation this partial derivative can be upwinded only along the x-direction following the path (x-u*dt). Therefore, there is no theoretical reason to produce a 2D stencil for this term.
The 2D stencil would appear if you consider the vector term v.Grad f and you consider the directional derivative df/ds along the vector direction (x -v*dt).

A further approach where a 2D stencil appear is if you use the integral form of the equation and you reconstruct the numerical flux to be integrated along the FV face.

You can see some upwinded schemes in Eq.s(25) in:
https://www.researchgate.net/publica...mes_simulation

Dear Sir,

Thank you for replying to all my queries. I do agree with you that I am not following any systematic approach.

About the second part of your reply,the scheme I am working on is an extension of local radial basis function in differential quadrature (LRBF-FD) method[1] for 2D convection-diffusion equation. This is a meshless/meshfree method and the domain is discretized using nodal/point cloud.
To approximate any differential operator at point x_i, a computational molecule is obtained by drawing a circle/sphere around that point with centre at x_i with a given radius, say, r_0. All the points inside the circle/sphere are called neighbors of x_i, and these neighbors are used to find weight values. Irrespective of what the differential operator is, all the neighbors are used in the calculating the weights.

One way to achieve upwinding in this scheme is to use a comet shape local support domain in which more upstream points compared to downstream points are used. This is done in reference 2.

I hope things make a little more sense now. Right Sir?

References:
[1]Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier–Stokes equations
https://www.sciencedirect.com/scienc...45782502006187
[2] A novel upwind-based local radial basis function differential quadrature method for convection-dominated flows - YL Chan
https://www.sciencedirect.com/scienc...45793013004155

[FMDenaro]

Originally Posted by alibaig1991

Dear Sir,

Thank you for replying to all my queries. I do agree with you that I am not following any systematic approach.

About the second part of your reply,the scheme I am working on is an extension of local radial basis function in differential quadrature (LRBF-FD) method[1] for 2D convection-diffusion equation. This is a meshless/meshfree method and the domain is discretized using nodal/point cloud.
To approximate any differential operator at point x_i, a computational molecule is obtained by drawing a circle/sphere around that point with centre at x_i with a given radius, say, r_0. All the points inside the circle/sphere are called neighbors of x_i, and these neighbors are used to find weight values. Irrespective of what the differential operator is, all the neighbors are used in the calculating the weights.

One way to achieve upwinding in this scheme is to use a comet shape local support domain in which more upstream points compared to downstream points are used. This is done in reference 2.

I hope things make a little more sense now. Right Sir?

References:
[1]Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier–Stokes equations
https://www.sciencedirect.com/scienc...45782502006187
[2] A novel upwind-based local radial basis function differential quadrature method for convection-dominated flows - YL Chan
https://www.sciencedirect.com/scienc...45793013004155

I don't use meshless methods and I am not an expert about them.
What I wonder is about the mathematical consistence of your discretization. Specifically, if you get your 2D stencil for the d/dx derivative, the local truncation error (I am using a terminology for mesh-based discretization) would contain derivatives also along y. Therefore, you should assess that for dx and dy going to zero (and I don't know how you could relate this to a meshless method in a rigorous way, I suspect by tending the radius to zero) the local truncation error vanishes.

However, considering the mathematical character of the convective equation, the upwinding is based on the characteristic lines idea. The informations travel along the trajectory dx/dt = v. Again, there is no mathematical reason to consider in the circonference of radius R around a point x the points outside the domain of dependence are used. What I would try to do is to upwind the center of the circumference.