Combining discontinuous Galerkin and finite volume methods within the Nmesh code

The Nmesh code is intended to allow for more efficient simulations of black holes and neutron stars, using a discontinuous Galerkin (DG) method. When using a DG method we usually employ a large number of touching computational domains or elements. The calculations in each element are then largely independent, and the only element-to-element communication needed, is between nearest neighbors and involves only boundary points. This means the method allows for very efficient parallelization. Such a DG method is particularly efficient when the evolved fields are smooth. Yet, in the case of shocks, that can occur e.g. in neutron stars, it is often better to use finite volume (FV) methods. However, in traditional FV methods one needs overlapping domains and more communication which can degrade parallelization efficiency. Here we describe a particular FV method that works with touching domains. We describe how this FV method functions, and how it can be combined with the DG method. We also show results from test cases such as scalar field evolutions and shock tube tests. In addition, we show some results where we evolve neutron stars.