Preface to the special issue “Recent Advances in Numerical Methods for Hyperbolic Partial Differential Equations”

摘要

Hyperbolic partial differential equations (PDE) are a very powerful mathematical tool to describe complex dynamic processes in science and engineering. Very often, hyperbolic PDE can be derived directly from first principles in physics, such as the conservation of mass, momentum and energy. These principles are universally accepted to be valid and can be used for the simulation of a very wide class of different problems, ranging from astrophysics (rotating gas clouds, the merger of a binary neutron star system into a black hole and the associated generation and propagation of gravitational waves) over geophysics (generation and propagation of seismic waves after an earthquake, landslides, avalanches, tidal waves, storm surges, flooding and morphodynamics of rivers) to engineering (turbulent flows over aircraft and the associated noise generation and propagation, rotating flows in turbo-machinery, multi-phase liquid-gas flows in internal combustion engines) and computational biology (blood flow in the human cardio-vascular system).It is of course impossible to cover all the above topics in a single special issue. However, all these apparently different applications have a common mathematical description under the form of nonlinear hyperbolic systems of partial differential equations, possibly containing also higher order derivative terms, non-conservative products and nonlinear (potentially stiff) source terms. From the mathematical point of view, the major difficulties in these systems arise due to the inherent nonlinearities and the formation of non-smooth solution features such as e.g. shock waves. The construction of robust and accurate numerical methods for such type of problems is even after decades of successful research an ongoing quest. This quest is fueled by recent advances in the development of novel methods with promising additional properties such as e.g. high spatial order on unstructured meshes, algorithmic simplicity for modern multi-core architectures and automatic mesh and/or trial function adaptation. The final goal is to construct methods that efficiently produce reliable results for such type of problems.This special issue is dedicated to recent advances in numerical methods for such nonlinear systems of hyperbolic PDE and tries to cover a wide spectrum of different problems and numerical approaches.