Numerical schemes for convection-dominated convection diffusion equation?

Question

We know that “convection dominant” will cause spurious oscillations in the solution. To avoid it, the most crude way is to apply upwind scheme. Is there any more sophisticated method? What’s their advantage? I heard discontinuous galerkin method may work

By the way, is there any other alternative name for “convection dominated problems”? I didn’t find too many information on it.

Answer 1

Even though I'm quite late to the party, my answer might help others with the same question. In general there are several discretization techniques for approximating the solution to convection-diffusion(-reaction) equations. As far of my knowledge, there are

1. Standard $P_k$ finite element methods with and without the SUPG stabilization. They are comparatively easy to implement, but usually produce large spurious oscillations in the convection dominated regime. A further drawback is that the SUPG parameter has to be chosen for which only bounds are known but not the precise value.
2. Another stabilization technique are so called spurious oscillations at layers diminishing (SOLD) methods. They produce usually less spurious oscillations than SUPG, but again a parameter has to be chosen of which even less is known compared to the SUPG parameter.
3. DG methods even with upwind discretization of the convective term have usually sharp layers but often still lack from spurious oscillations. These oscillations might be smaller compared to SUPG but they are still considerable. Slope limiter are a post-processing technique to reduce these oscillations further. If I remember correctly, there exist also SUPG variants of the DG method, but I don't have a citation at hand.
4. There are of course finite volume techniques that produce results without any oscillations. But they are lowest-order ($P_0$) only and the layer is smeared.
5. Another technique is called algebraic flux correction (AFC) methods. They are constructed such that spurious oscillations cannot occur, but they also have the drawback of being lowest-order only ($P_1$) only and it is a non-linear method. So it can be potentially expensive but qualitatively it produces the best solutions.

A comparison of these techniques except for the AFC method can be found in this publication. In general the group of Volker John, which I am also a part of, does a lot on convection-dominated convection-diffusion-reaction equations. But there are also other people working in this field, e.g., Petr Knobloch and Gabriel Barrenechea.

I hope this help :)

Answer 2

You might have some luck searching for methods for advection-diffusion equations and looking at methods that specifically handle the advection term with more care. As for methods, Discontinuous Galerkin was developed for these types of problems, so it is a natural fit. Finite Volume Methods were also designed with advection problems in mind and there are relatively simple adjustments that can account for diffusion. FVM has the benefit that many of its schemes can be easily recast in terms of finite difference methods, which are easy to interpret.

Book on FVM by LeVeque Book on DG by Dolejší, Vít, Feistauer, and Miloslav