I am conducting compressible flow simulations based on the modal discontinuous Galerkin method. In doing this, I sometime found that there is another version of DG method, the nodal DG one. Is there anyone knowing the difference between two versions and advantages/disadvantages of each method? I wonder if there is a big difference. For compressible flow simulations, which one is better?

can you send your modal DG result picture to me? dirac_euler@hotmail.com

nodal DG uses lagrange interploration so it is difficult to deal with boundary node, it belong to H refinement,while modal DG is something like spectral method,it can get a p refinement.

With DG, each element is independent so either method can be used with h- and p-refinement. They are pretty much equivalent at low order, but their computational complexity is potentially different at high order. This applies to both the DG and continuous Galerkin.

If the problem is linear, then a clever choice of basis can make the modal stiffness matrix sparse. This is not so for nonlinear problems or for a nodal basis. However, a nodal basis is well suited to Q1 finite element preconditioning. That is, discretize the Jacobian on the nodes of the high order basis using Q1 finite elements. This approximation to the real Jacobian is very sparse and spectrally equivalent. In a (Jacobian-Free) Newton-Krylov iteration, you need only assemble this preconditioner. Thus constructing the preconditioner is a cheap as for Q1 finite elements, but the high-order method converges in a constant number of iterations.

The key choice is whether you want a hierarchical basis (for modal multigrid preconditioning) or a sparse preconditioning matrix to which you can apply standard preconditioners. If you are not exploiting these properties, then modal vs. nodal doesn't really matter.

Spectral equivalence has now been proven: @article{kim2007pbp, title={{Piecewise bilinear preconditioning of high-order finite element methods}}, author={Kim, S.D.}, journal={Electronic Transactions on Numerical Analysis}, volume={26}, pages={228--242}, year={2007} }

[cfdnewbie]

Just to add a bit to the discussion: the fastest existing DG is a spectral, nodal DG on hexas (see e.g. book by Kopriva: implementing spectral methods). It uses interpolation as integration points, and thus resembles collocation spectral methods closely in terms of speed.

[kielman]

If you are still interested in this question you should read this thread :
http://scicomp.stackexchange.com/que...-disadvantages