Thank you for responding Wolfgang Bangerth.
The GCL condition comes from the discretized scheme satisfying free-stream
preservation. I will demonstrate this for 2D below, (can be interpreted for
spectral, DG, finite difference, finite volume etc):
Consider the conservation law: \frac{\partial W}{\partial t} +
\frac{\partial F}{\partial x} +\frac{\partial G}{\partial y} =0
Transforming this to the reference computational space (x,y)->(\xi, \eta):
J*\frac{\partial W}{\partial t} + J*\frac{ \partial \xi}{\partial x} *
\frac{\partial F}{\partial \xi} + J * \frac{ \partial \eta}{\partial x}*
\frac{\partial F}{\partial \eta} + J * \frac{ \partial \xi}{\partial y} *
\frac{\partial G}{\partial \xi} + J*\frac{ \partial \eta}{\partial
y}*\frac{\partial G}{\partial \eta}
Putting this in conservative form results in:
J\frac{\partial W}{\partial t} + \frac{\partial}{\partial \xi} (
J*F*\frac{\partial \xi}{\partial x} +J*G*\frac{\partial \xi}{\partial y} )
+ \frac{\partial}{\partial \eta} ( J*F*\frac{\partial \eta}{\partial x}
+J*G*\frac{\partial \eta}{\partial y} ) - F*( GCL in x) - G*(GCL in y) =0
where GCL in x = \frac{\partial }{\partial \xi} ( det(J)* \frac{\partial
\xi
}{\partial x}) + \frac{\partial }{\partial \eta}( det(J)* \frac{\partial
\eta}{\partial x} )
similarly for y.
So for the conservative numerical scheme to satisfy free stream
preservation, the GCL conditions must go to zero.
For linear grids, there are no issues with the classical definition for the
inverse of the Jacobian, but what Kopriva had shown (before him Thomas and
Lombard), was that the metric Jacobian has to be calculated in either a
"conservative curl form" or an "invariant curl form" since it reduces the
GCL condition to the divergence of a curl, which is always discretely
satisfied. In the paper by Kopriva, he shows this, an example in 3D:
Analytically
J*\frac{\partial \xi}{\partial x} = \frac{\partial z}{\partial \zeta} *
\frac{\partial y}{\partial \eta} - \frac{\partial z}{\partial \eta} *
\frac{\partial y}{\partial \zeta}
but the primer doesn't satisfy free-stream preservation while the latter
("conservative curl form") does.
I will put together a unit test for a curvilinear grid.
Thank you,
Alex
On Tuesday, June 16, 2020 at 10:24:59 PM UTC-4, Wolfgang Bangerth wrote:
>
>
> Alexander,
>
> > I am wondering if anybody has also found that the inverse of the
> Jacobian from
> > FE Values, with MappingQGeneric does not satisfy the Geometric
> Conservation
> > Law (GCL), in the sense of:
> >
> > Kopriva, David A. "Metric identities and the discontinuous spectral
> element
> > method on curvilinear meshes." /Journal of Scientific Computing/ 26.3
> (2006): 301.
> >
> > on curvilinear elements/manifolds in 3D.
> > That is:
> > \frac{\partial }{\partial \hat{x}_1} *det(J)* \frac{\partial \hat{x}_1
> > }{\partial x_1} + \frac{\partial }{\partial \hat{x}_2} *det(J)*
> \frac{\partial
> > \hat{x}_2}{\partial x} + \frac{\partial }{\partial \hat{x}_3} *
> > det(J)*\frac{\partial \hat{x}_3 }{\partial x_1} != 0 (GCL says it should
> =0,
> > similarly for x_2 and x_3)
> >
> > If so or if not, also, has anybody found a remedy to have the inverse of
> the
> > Jacobian from FE Values with MappingQGeneric to satisfy the GCL.
>
> I'm not sure any of us have ever thought about it. (I haven't -- but I
> really
> shouldn't speak for anyone else.) Can you explain what this equality
> represents? Why should it hold?
>
> I'm also unsure whether we've ever checked whether it holds (exactly or
> approximately). Can you create a small test program that illustrates the
> behavior you are seeing?
>
> Best
> W.
>
> --
> ------------------------------------------------------------------------
> Wolfgang Bangerth email: bang...@colostate.edu
> <javascript:>
> www: http://www.math.colostate.edu/~bangerth/
>
>
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