I have a couple of questions with regards to the usage of flux/slope limiters, which I was hoping to get some input. I apologise in advance if my questions are trivial, but flux limiters lie way beyond my research area (up until now) and I am still trying to make sense of all information that is out there.

I am trying to implement a slope limiter in my code in an attempt to smooth some of the oscillations that are present in my solution*, however due to the complex nature of my discretised PDE (the Neutron Transport Equation) it can be very expensive (at best $$O(n^2)$$ ) to project from the discretised solution to the physical solution i.e. solution evaluated on the quadrature points and then apply the limiter.

From what I have encountered in the literature people usually first project to the quadrature points before applying the limiter, however I have encountered a few examples where that is not the case, for instance one paper:

Limiting is done by comparing the values of Ui(x) to the solution averages on neighboring elements, where the points x can be quadrature points, element vertices, edge midpoints, or other. [1]

Question

So my question is, would it be correct to apply the limiter on my discretised solution i.e. coefficients of a series expansion, in this case a real Spherical Harmonics or a Wavelet series instead of my solution projected on the quad points? (coefficients of both expansions can naturally be negative)

And if so, would I have to use a specific type of limiters or would a Van-Leer, minmod, etc. be fine?

1. $$\mathbf{solution^1}$$: the field I want to limit is not actually the solution, it is a field that is associated with the solution and holds information about the solution's error. The actual PDE solution and consequently the underlying discretisation, are stable and produce non-oscillatory solutions. It is this surrogate field that is causing me grief.

2. elements used: first order triangular (2D) and tets (3D)

3. discretisation is not DG FEM, so no upwinding

4. PDE is linear first order if we consider the simple monoenergetic case

Spatial discretisation

The underlying underlying FEM discretisation that introduces the discontinuity and needs to be limited is not a simple DG so we don't have any upwinding, here is the paper where the method is described in case anyone is interested: https://doi.org/10.13182/NSE08-82