Review of Summation-by-parts schemes for initial-boundary-value problems

High-order finite difference methods are efficient, easy to program, scales well in multiple dimensions and can be modified locally for various reasons (such as shock treatment for example). The main drawback have been the complicated and sometimes even mysterious stability treatment at boundaries and interfaces required for a stable scheme. The research on summation-by-parts operators and weak boundary conditions during the last 20 years have removed this drawback and now reached a mature state. It is now possible to construct stable and high order accurate multi-block finite difference schemes in a systematic building-block-like manner. In this paper we will review this development, point out the main contributions and speculate about the next lines of research in this area

Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws

The development of reliable numerical methods for the simulation of real life problems requires both a fundamental knowledge in the field of numerical analysis and a proper experience in practical applications as well as their mathematical modeling. Thus, the purpose of the workshop was to bring together experts not only from the field of applied mathematics but also from civil and mechanical engineering working in the area of modern high order methods for the solution of partial differential equations or even approximation theory necessary to improve the accuracy as well as robustness of numerical algorithms

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 error-based step size control for discontinuous Galerkin methods for compressible fluid dynamics

We study temporal step size control of explicit Runge-Kutta methods for compressible computational fluid dynamics (CFD), including the Navier-Stokes equations and hyperbolic systems of conservation laws such as the Euler equations. We demonstrate that error-based approaches are convenient in a wide range of applications and compare them to more classical step size control based on a Courant-Friedrichs-Lewy (CFL) number. Our numerical examples show that error-based step size control is easy to use, robust, and efficient, e.g., for (initial) transient periods, complex geometries, nonlinear shock capturing approaches, and schemes that use nonlinear entropy projections. We demonstrate these properties for problems ranging from well-understood academic test cases to industrially relevant large-scale computations with two disjoint code bases, the open source Julia packages Trixi.jl with OrdinaryDiffEq.jl and the C/Fortran code SSDC based on PETSc