Abstract
In this work we follow the methodology of higher order finite volume (FV) and spectral volume (SV) schemes and introduce a reconstruction operator into the discontinuous Galerkin (DG) method. This operator constructs higher order piecewise polynomial reconstructions from the lower order DG scheme.
This allows us to increase the accuracy of existing DG schemes with a problem-independent reconstruction procedure. Unlike the FVM, the reconstruction stencil has minimal size independent of the approximation order.
Such a procedure was already proposed based on heuristic arguments, however we provide a more rigorous derivation, which justifies the increased order of accuracy. Furthermore, we provide an alternative construction of the reconstruction procedure based on the SV method.
Numerical experiments are carried out.