A free-energy stable nodal discontinuous Galerkin approximation with summation-by-parts property for the Cahn-Hilliard equation

We present a nodal Discontinuous Galerkin (DG) scheme for the Cahn-Hilliard equation that satisfies the summation-by-parts simultaneous-approximation-term (SBP-SAT) property. The latter permits us to show that the discrete free-energy is bounded, and as a result, the scheme is provably stable. The scheme and the stability proof are presented for general curvilinear three-dimensional hexahedral meshes. We use the Bassi-Rebay 1 (BR1) scheme to compute interface fluxes, and an IMplicit-EXplicit (IMEX) scheme to integrate in time. Lastly, we test the theoretical findings numerically and present examples for two and three-dimensional problems

Entropy Stable Summation-by-Parts Methods for Hyperbolic Conservation Laws on h/p Non-Conforming Meshes

In this work we present high-order primary conservative and entropy stable schemes for hyperbolic systems of conservation laws with geometric (h) and algebraic (p) non-conforming rectangular meshes. Throughout we rely on summation-by-parts (SBP) operators which discretely mimic the integration-by-parts rule to construct stable approximations. Thus, the discrete proofs of primary conservation and entropy stability can be done in a one-to-one fashion to the continuous analysis. Here, we consider different SBP operators based on finite difference as well as discontinuous Galerkin approaches. We derive non-conforming schemes by extending ideas of high-order primary conservative and entropy stable SBP methods on conforming meshes. Here, special attention is given to the coupling between non-conforming elements. The coupling is instructed to entropy stable projection operators. However, these projection operators suffer from a suboptimal degree. Therefore, we develop degree preserving SBP operators where the norm matrix has a higher degree compared to classical SBP operators. With these operators it is possible to construct entropy stable projection operators which have the same degree as the SBP differentiationmatrix. Typically, high-order primary conservative and entropy stable schemes are semi-discrete methods with a discretized spatial domain and assuming continuity in time. Therefore, temporal errors are introduced when integrating the conservation laws in time with standard methods, e.g. Runge-Kutta schemes, for which the entropy can have an unpredictable temporal behaviour. Thus, we extend high-order primary conservative and entropy stable semi-discrete methods to fully-discrete schemes on conforming and non-conforming meshes. This results in an implicit space-time method. We introduce a simple mesh generation strategy to obtain quadrilateral meshes surrounding two dimensional complex geometries. Finally, with the generated meshes we simulate a flow around a NACA0012 airfoil to demonstrate the benefits of considering non-conforming elements for a practical simulation

On discretely entropy conservative and entropy stable discontinuous Galerkin methods

High order methods based on diagonal-norm summation by parts operators can be shown to satisfy a discrete conservation or dissipation of entropy for nonlinear systems of hyperbolic PDEs. These methods can also be interpreted as nodal discontinuous Galerkin methods with diagonal mass matrices. In this work, we describe how use flux differencing, quadrature-based projections, and SBP-like operators to construct discretely entropy conservative schemes for DG methods under more arbitrary choices of volume and surface quadrature rules. The resulting methods are semi-discretely entropy conservative or entropy stable with respect to the volume quadrature rule used. Numerical experiments confirm the stability and high order accuracy of the proposed methods for the compressible Euler equations in one and two dimensions