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