[hunt_mat]

How good are interpolating polynomials in CFD?

I'm working on a finite volume code, and I need to find the value of the density on the boundary and I only have the midpoint values. I thought about using the first three mid-point values to obtain a quadratic equation rho(x)=ax^2+bx+c, and then set x=0 to obtain the value at the boundary.

Does this sound reasonable?

[FMDenaro]

Originally Posted by hunt_mat

How good are interpolating polynomials in CFD?

I'm working on a finite volume code, and I need to find the value of the density on the boundary and I only have the midpoint values. I thought about using the first three mid-point values to obtain a quadratic equation rho(x)=ax^2+bx+c, and then set x=0 to obtain the value at the boundary.

Does this sound reasonable?

But that would be an extrapolation…
I used to compute the value on a fiinite volume surrounding the node on the boundary.

[sbaffini]

I would start from what you know on the boundary. If density is involved in your calculation (i.e., it isn't constant), and you don't already know it in a point, I expect you to know some other variable from which density can be desumed, like pressure and temperature.

Unless you are actually working out how to apply your boundary conditions in the first place... in that case, as mentioned by Filippo, do not extrapolate at higher order, ever. In FV codes, if some variable needs to be extrapolated, it is typically copied from the nearest cell center. But it depends from which boundary condition you are working on and the kind of code.

You need to tell more (kind of code, kind of bc, additional details on the boundary and scope of the task, etc.) in order to have a sharper suggestion.

[hunt_mat]

Originally Posted by FMDenaro

But that would be an extrapolation…
I used to compute the value on a fiinite volume surrounding the node on the boundary.

This doesn't actually tell be anything. I've tried that method before, and it failed.

[FMDenaro]

Originally Posted by hunt_mat

This doesn't actually tell be anything. I've tried that method before, and it failed.

Describe the way you tried. I built a volume surrounding the node on the wall and computed the fluxes on the faces in the flow interior (no mass flux on the wall, of course)-

[hunt_mat]

Originally Posted by sbaffini

I would start from what you know on the boundary. If density is involved in your calculation (i.e., it isn't constant), and you don't already know it in a point, I expect you to know some other variable from which density can be desumed, like pressure and temperature.

I assume you're referring to some kind of constitutive equation, like an ideal gas. I don't have that unfortunately.

Unless you are actually working out how to apply your boundary conditions in the first place... in that case, as mentioned by Filippo, do not extrapolate at higher order, ever. In FV codes, if some variable needs to be extrapolated, it is typically copied from the nearest cell center. But it depends from which boundary condition you are working on and the kind of code.

I've heard this from someone, can you tell me why?

You need to tell more (kind of code, kind of bc, additional details on the boundary and scope of the task, etc.) in order to have a sharper suggestion.

I'm solving the set of PDEs:
v_t=u_h, u_t=(P_L(v)+zeta(v)/v*u_h)_h

Where v is specific volume and h denotes the amount of mass from x=0. I have stress-free conditions a P_L(v) and zeta(v) are functions of v.

[hunt_mat]

This is the method I used.

[FMDenaro]

Originally Posted by hunt_mat

This is the method I used.

If I understand, you can write Eq.(8) on the boundary and solve it

[hunt_mat]

Originally Posted by FMDenaro

If I understand, you can write Eq.(8) on the boundary and solve it

I need to find the u_i on the cell interfaces. That implies I need to find them on the boundary. I thought that the extrapolation I did provided the appropriate extrapolation, or so I thought.

[sbaffini]

I am not accustomed to this kind of problem but, I can only notice that:

1) You don't actually need a bc on v, you need v on the boundary because it is needed by the bc on u. Indeed, the volume integrated equation for v only has u as flux terms. If you have u on boundaries you can evolve v in any cell, it seems to me. Is this interpretation correct?

2) I know nothing about sintering, but not having a bc is weird. I would look better at some literature on the matter to find out how bcs are applied here.

3) In any case, if there isn't a bc, extrapolating the nearest cell value is just as good as using the second order interpolation, but it has the added benefit of not introducing any new maxima or minima. Introducing new maxima or minima trough bcs is an almost certain way to divergence (altough, there might be cases where this doesn't happen).

[hunt_mat]

Originally Posted by sbaffini

I am not accustomed to this kind of problem but, I can only notice that:

1) You don't actually need a bc on v, you need v on the boundary because it is needed by the bc on u. Indeed, the volume integrated equation for v only has u as flux terms. If you have u on boundaries you can evolve v in any cell, it seems to me. Is this interpretation correct?

The boundary condition for u on the boundary comes in the form of a Neumann condition, so you have du/dh on the boundary, this I used for the boundary conditions to get the u on the boundary from the stress free condition. This requires the information on v on the boundary

2) I know nothing about sintering, but not having a bc is weird. I would look better at some literature on the matter to find out how bcs are applied here.

I have no idea what the boundary condition might be. All I know is that the total mass remains the same. I'm not sure what to use as a boundary condition. I could say that the mass flux through the boundary is zero, that will lead to dv/dh=0 on each boundary.

3) In any case, if there isn't a bc, extrapolating the nearest cell value is just as good as using the second order interpolation, but it has the added benefit of not introducing any new maxima or minima. Introducing new maxima or minima trough bcs is an almost certain way to divergence (altough, there might be cases where this doesn't happen).

So I don't need to use my approach by fitting a quadratic?