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There are two general approaches to representing solutions in the discontinuous galerkin method: nodal and modal.

  1. 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.

  2. 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.

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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.

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  • $\begingroup$ Thanks, good points. Any insight into quadrature/integration, and the implementation of limiters for discontinuities in the two approaches? $\endgroup$ – Aurelius Nov 23 '13 at 14:16
  • $\begingroup$ Modal and nodal basis functions are usually designed to span the same space. I added a note about collocated quadrature. No linear high-order basis can capture extrema directly, so implementation of limiters is very similar for the methods I am familiar with. $\endgroup$ – Jed Brown Nov 23 '13 at 19:21
  • $\begingroup$ Thanks again, accepting this answer. One last subjective question: if you were starting a new general-purpose CFD project leveraging something like petsc, would you have a strong preference for nodal vs modal? $\endgroup$ – Aurelius Dec 3 '13 at 18:47
  • $\begingroup$ I think nodal methods are almost always more practical. The operations that are "more elegant" for modal bases are not bottlenecks. $\endgroup$ – Jed Brown Dec 3 '13 at 19:58
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I was curious to see some answers to this question, but somehow nobody bothers to reply...

Regarding literature, I really like the book Spectral/hp Element Methods for Computational Fluid Dynamics (there's also a cheaper soft-cover version now) and also the book of Hesthaven and Warburton. These two go into quite some detail that will help you implement the methods. The book of Canuto, Hussaini, Quarteroni and Zang is more theoretical. This one also has a second volume "Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics".

I don't work on DG methods and I'm not an expert to judge the advantages of nodal vs. modal. The book of Karniadakis & Sherwin is more focused on methods with continuous modal expansions. In this type of method, you are obliged to reorder the modes in two neighbouring elements in such fashion that the corresponding modes on the interface match in order to preserve the continuity of the global expansion. In addition, imposing boundary conditions requires extra attention since your modes are not associated with a specific location on the boundary.

I hope someone familiar with this type of methods will add more detail.

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  • $\begingroup$ Thanks, I'm bummed I haven't gotten any good answers here too! I have both the Karniadakis/Sherwin (modal & continuous) and Hesthaven/Warburton (nodal) books and I also recommend them. I'm comfortable with the implementations, it's just the pros/cons that aren't clear to me. $\endgroup$ – Aurelius Nov 21 '13 at 16:06

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