There are two general approaches to representing solutions in the discontinuous galerkin method: nodal and modal.Modal: Solutions are represented by sums of modal coefficients multiplied by a set of polynomials, e.g. $u(x,t) = \sum_{i=1}^N u_i(t) \phi_i(x)$ where $\phi_i$ is usually orthogonal polynomials, e.g. Legendre. One advantage of this is that the orthogonal polynomials generate a diagonal mass matrix.Nodal: Cells are comprised of multiple nodes on which the solution is defined. Reconstruction of the cell is then based on fitting an interpolating polynomial, e.g. $u(x,t) = \sum_{i=1}^N u_i(x,t) l_i(x)$ where $l_i$ is a Lagrange polynomial. One advantage of this is that you can position your nodes at quadrature points and quickly evaluate integrals.In the context of a large-scale, complex ($10^6$-$10^9$ DOFs) 3D mixed structured/unstructured parallel application with goals of flexibility, clarity of implementation, and efficiency, what are the comparative advantages and disadvantages of each method?I'm sure there's good literature already out there, so if someone could point me to something that'd be great as well.
Discontinuous Galerkin: Nodal vs Modal advantages and disadvantages
fluid dynamics;discontinuous galerkin
The tradeoffs below apply equally to DG and to spectral elements (or $p$-version finite elements).Changing the order of an element, as in $p$-adaptivity, is simpler for modal bases because the existing basis functions do not change.  This is generally not relevant to performance, but some people like it anyway.  Modal bases can also be filtered directly for some anti-aliasing techniques, but that is also not a performance bottleneck.  Modal bases can also be chosen to expose sparsity within an element for special operators (usually the Laplacian and mass matrices).  This does not apply to variable coefficient or non-affine elements, and the savings are not huge for the modest order typically used in 3D.Nodal bases simplify the definition of element continuity, simplify implementation of boundary conditions, contact, and the like, are easier to plot, and lead to better $h$-ellipticity in discretized operators (thus allowing use of less expensive smoothers/preconditioners).  It is also simpler to define concepts that are used by solvers, such as rigid body modes (just use nodal coordinates), and to define certain grid transfer operators such as arise in multigrid methods.  Embedded discretizations are also readily available for preconditioning, without needing a change of basis.  Nodal discretizations can efficiently use collocated quadrature (as with spectral element methods), and the corresponding under-integration can be good for energy conservation.  Inter-element coupling for first-order equations is sparser for nodal bases, though otherwise-modal bases are often modified to obtain the same sparsity.