On structure-preserving and pointwise conservative continuous DG schemes for hyperbolic systems
We present a new class of structure-preserving semi-discrete continuous-discontinuous Galerkin (CG-DG) finite element schemes for linear and nonlinear hyperbolic systems of partial differential equations on unstructured simplex meshes that automatically satisfy the following properties: i) the new schemes are not only cellwise conservative, but also locally pointwise conservative everywhere, hence they satisfy the integral form of the conservation law on arbitrary control volumes that do not have to coincide with the mesh at all; ii) the new methods naturally satisfy the two basic vector calculus identities $\nabla \cdot \nabla \times \mathbf{A}$ and $\nabla \times \nabla Z$ exactly pointwise locally and globally everywhere on the discrete level; iii) for linear symmetric hyperbolic systems the schemes are naturally energy conservative for the square energy, i.e. nonlinearly stable in the $L^2$ norm. The key ingredient of the new CG-DG schemes is the use of two different but compatible approximation spaces: the classical DG space $\mathcal{U}_h^N$ of discontinuous piecewise polynomials of degree up to $N$ and a classical finite element space $\mathcal{W}_h^{N+1}$ of globally continuous piecewise polynomials of degree $N+1$. In the new CG-DG schemes, the discrete solution $\mathbf{u}_h$ is sought in $\mathcal{U}_h^N$, while a suitable discrete flux field $\tilde{\mathbf{f}}_h$ is computed in $\mathcal{W}_h^{N+1}$. For $N=0$ our new schemes are directly related to cell-centered finite volume schemes with suitable vertex-based fluxes. All claimed properties of the schemes are first mathematically proven and are then also verified via suitable numerical tests. We show applications of our approach to three linear and nonlinear hyperbolic systems.