[dune-pdelab] Finite Volume Method with slope reconstruction

[Marian Piatkowski]

Hi Gregor


On 04/28/2017 03:13 PM, Gregor Corbin wrote:
> Hello dear Dune developers,
>
> I have a dune-pdelab code that implements a first-order finite volume 
> method for a hyperbolic system. Currently I am trying to implement a 
> slope reconstruction to achieve second order accuracy in space. 
> Therefore, before each stage of the time-stepping scheme, I need to 
> iterate over the grid and compute the slopes on each cell.
> My current approach is to write a limiter class and give it to the 
> explicit one step method like this:
>
> oneStepMethod.apply(t,dt,x_old,x_new,LinearReconstruction);

That's how it is exactly done in one of our Dune modules for the 
lecture, dune-funcep.
Maybe you could have a look at the code in our Git repository
- https://parcomp-git.iwr.uni-heidelberg.de/Teaching/dune-funcep
especially the file
- 
https://parcomp-git.iwr.uni-heidelberg.de/Teaching/dune-funcep/blob/master/src/common/limiter.hh

I think the second class in this file is just what you are looking for.

Regards,

Marian

>
> In the prestage method of this class, which gets only the solution 
> vector, i iterate over all the cells on the grid. For each cell I want 
> to extract the solution on that cell and on it's neighbors.
> But I have no idea how to do this.
>
> I appreciate any help on this problem. Is there maybe even a better 
> way to implement slope limiting in pdelab?
>
> Cheers and thanks in advance,
> Gregor Corbin
>
> Below is a sketch for the code of the limiter class:
>
> template <typename GridView, typename GFS, int n_components>
> class LinearReconstruction
> {
>   enum {dim = GridView::dimension};
>   enum {n_comp = n_components};
>
> public:
>   LinearReconstruction(const GridView &grid_view, const GFS &gfs):
>   grid_view_(grid_view),
>   gfs_(gfs),
>   mapper_(grid_view_),
>   slopes_(mapper_.size(),0.0)
>   {}
>
>   template<typename X>
>   void prestage(X& x)
>   {
>     // reconstruct slopes
>     for (const auto& cell : elements(grid_view_) )
>     {
>        // I would like to do something like in the local operator 
> methods,
>        // but how do I get the lfsu_s ???
>        // is there a method to get it from the cell and the grid 
> function space?
>
>        // lfsu_s = gfs_.localFunctionSpace(cell); <- does something 
> similar exist?
>       Dune::FieldVector<double,n_comp> u_s(0.0);
>       for (int k = 0; k < n_comp; ++k)
>         u_s[k] = x(lfsu_s.child(k),0);
>
>       // get the solutions from all the neighbors
>       std::vector<Dune::FieldVector<double,n_comp> > 
> u_n(pow(2,dim),0.0); // only valid for structured cubic grids
>       int i = 0;
>       for (const auto& is : intersections(grid_view_,cell))
>       {
>         const auto &neighbor = is.outside();
>          // lfsu_n = gfs_.localFunctionSpace(neighbor); <- does 
> something similar exist?
>
>          for (int k = 0; k < n_comp; ++k)
>               u_n.at(i)[k] = x(lfsu_n.child(k),0);
>          ++i;
>       }
>
>       // use the info to reconstruct slopes
>       slopes_.at(mapper_.index(cell)) =  // some function(u_s, u_n)
>     }
>   }
>
>   template<typename V>
>   void poststage(V& v)
>   {}
> private:
>   const GridView &grid_view_;
>    const GFS &gfs_;
>   using Mapper =
> Dune::MultipleCodimMultipleGeomTypeMapper<GridView,Dune::MCMGElementLayout>; 
>
>   const Mapper mapper_;
>   std::vector<Dune::FieldMatrix<double,n_comp,dim> > slopes_;
> };
>
>
>
>
>
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