Interesting... In my experience dG works without stabilization and additional artificial viscosity in convection-dominated incompressible flows (let's consider incompressible flows to avoid issues with discontinuities and monotonicity). One interesting point is that some formulations that works very well in practice are not conservative since the convective term is not written in divergence form. E.g. the skew-symmetric form introduced by Temam in combination with centered fluxes works perfectly. With this choice the energy is conserved (even if the velocity field is only weekly divergence free), but not the momentum.
In general I think that discretizations based on numerical fluxes are more robust, in particular when the fluxes are based on the physics. Relaxing the continuity requirements might help in case of insufficient spatial resolution and week imposition of boundary conditions also helps. I admit that my experience with supg is limited, I'm not able to "compare" with dG. Lorenzo On Apr 18, 2013 7:09 PM, "Jed Brown" <[email protected]> wrote: > Manav Bhatia <[email protected]> writes: > > -- How critical is this lack of conservation properties for general > > application of CFD for transonic and supersonic aerodynamic analyses? > > Ah, a question for the ages. A lot of CFD engineers will say that local > conservation is essential and they won't even consider a method that > isn't. Most commercial CFD packages use finite volume methods for this > reason. > > At the other end of the spectrum are the FOSLS advocates that say the > engineers just don't understand what they want and insist that > conservation always sacrifices something else. Other least squares > advocates (e.g. Pavel Bochev) insist that conservation is essential, but > that you should get it by using compatible spaces in the LS formulation > (erasing some notable benefits of LS in the process). > > In the end, it depends on the problem. > > > -- Is this apparant non-ideal conservation behavior a primary reason > > for interest in DG vs continuous FEM? > > Yes, conservation and stability. Linear DG satisfies a cell-wise > entropy inequality so that it can often be used without limiting. > Strict monotonicity is not practically available for continuous FEM. > > > -- The typical DG methods that I have seen do not add a stabilization > > within the domain, but handle the boundary terms to ensure continuity of > > flux across the discontinuous elements. However, a straight-forward > > application of Galerkin method is known to be unstable. So, I am assuming > > that the stabilization and non-oscillatory behavior comes from forgoing > the > > requirement of continuity and the treatment of boundary terms. However, > > would this also not imply that if one continued to increase the element > > size while increasing the polynomial order, the solution within the will > at > > some point become unstable/oscillatory? > > Yes, DG stability is only cell-wise. Normal flux limiting reduces the > method to second order even in smooth regions. So if you go to > high-order, you need more exotic limiting to prevent internal > oscillations (see Zhang and Shu 2010 and 2011). > > > ------------------------------------------------------------------------------ > Precog is a next-generation analytics platform capable of advanced > analytics on semi-structured data. The platform includes APIs for building > apps and a phenomenal toolset for data science. Developers can use > our toolset for easy data analysis & visualization. Get a free account! > http://www2.precog.com/precogplatform/slashdotnewsletter > _______________________________________________ > Libmesh-users mailing list > [email protected] > https://lists.sourceforge.net/lists/listinfo/libmesh-users > ------------------------------------------------------------------------------ Precog is a next-generation analytics platform capable of advanced analytics on semi-structured data. The platform includes APIs for building apps and a phenomenal toolset for data science. Developers can use our toolset for easy data analysis & visualization. Get a free account! http://www2.precog.com/precogplatform/slashdotnewsletter _______________________________________________ Libmesh-users mailing list [email protected] https://lists.sourceforge.net/lists/listinfo/libmesh-users
