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The other day, my computational fluid dynamics instructor was absent and he sent in his PhD candidate to substitute for him. In the lecture he gave, he seemed to indicate several disadvantages associated with various discretization schemes for fluid flow simulations:

Finite Difference Method: It is difficult to satisfy conservation and to apply for irregular geometries

Finite Volume Method: It tends to be biased toward edges and one-dimensional physics.

Finite Element Method: It is difficult to solve hyperbolic equations using FEM.

Discontinuous Galerkin: It is the best (and worst) of all worlds.

Fluctuation Splitting: They are not yet widely applicable.

After the lecture, I tried asking him where he got this information but he did not specify any source. I also tried to get him to clarify what he meant by DG being the "best and worst of all worlds", but couldn't get a clear answer. I can only assume that he came to these conclusions from his own experience.

From my own experience, I can only verify the first claim that FDM is difficult to apply to irregular geometries. For all other claims, I don't have sufficient experience to verify them. I'm curious how accurate these claimed 'disadvantages' are for CFD simulations in general.

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The proposed characteristics are reasonable in the sense that they roughly represent popular opinion. This question has massive scope, so I'll just make a few observations now. I can elaborate in response to comments. For more detailed related discussion, see What are criteria to choose between finite-differences and finite-elements?

  • Low order conservative finite difference methods are readily available for unstructured grids. High order non-oscillatory FD methods are another matter. In Finite Difference WENO schemes, the physics appears in a flux splitting that is not available for all Riemann solvers.

  • Finite volume methods work fine in multiple dimensions, but to go higher than second order for general flow structures, you need extra face quadrature points and/or transverse Riemann solves, greatly increasing the cost relative to FD methods. However, these FV methods can be applied to non-smooth and unstructured meshes and can use arbitrary Riemann solvers.

  • Continuous finite element methods can be used for CFD, but stabilization becomes delicate. It is not usually practical to have strictly non-oscillatory methods and stabilization often needs additional information like entropy. When the consistent mass matrix is used, explicit time stepping becomes much more expensive. Continuous Galerkin methods are not locally conservative, which causes problems for strong shocks. See also Why is local conservation important when solving PDEs?

  • Discontinuous Galerkin methods can use any Riemann solver to connect elements. They have better inherent nonlinear stability properties than the other common methods. DG is also rather complicated to implement and is not generally monotone inside an element. There are limiters for DG that ensure positivity or a maximum principle.

  • There are other methods like Spectral Difference (e.g. Wang et al 2007 or Liang et al 2009) that have the potential to be very efficient (like Finite Difference), while having more geometric flexibility and high order accuracy.

High Reynolds number flows have thin boundary layers, requiring highly anisotropic elements to solve efficiently. For incompressible or nearly incompressible elements, this causes significant trouble for many discretizations. For additional discussion, mostly from the perspective of finite element methods, see What spatial discretizations work for incompressible flow with anisotropic boundary meshes?

For steady problems, the ability to efficiently use nonlinear multigrid (FAS) is attractive. FD, FV, and DG methods can generally use FAS efficiently because, roughly speaking,

$$ \frac{(\text{cost per pointwise residual}) \cdot (\text{number of points})}{\text{cost of global residual}} \lesssim 2 . $$

This ratio is often more than 10 for continuous finite element methods. This ratio is not sufficient for efficient FAS with pointwise or elementwise smoothers, however. It is also necessary to have an $h$-elliptic discretization to use for defect correction, or otherwise modify the multigrid cycle. For further discussion, see Is there a multigrid algorithm that solves Neumann problems and has a convergence rate independent of the number of levels? A positive answer to this research question would potentially offer an efficient FAS for continuous finite elements.

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    $\begingroup$ Could you please add a reference that explains the idea of the spectral difference method? $\endgroup$ – shuhalo May 27 '12 at 14:31
  • $\begingroup$ Added references. I linked by DOI, but you can find author copies if you search. $\endgroup$ – Jed Brown May 27 '12 at 16:31
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In short for DG:

A consequence of relaxing the continuity requirements across element boundaries is that number of variables in DG-FEM is larger than for continuous counterpart for the same number of elements.

On the other hand because of local formulation (in terms of elements) we have following advantages:

  • Non-stationary and source terms are fully decoupled between elements. Mass matrices can be inverted at element level.
  • Easier parallelization.
  • Adaptive refinements (h-, p- and hp) are made easy - there is no need for global node renumbering.
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