Well-balanced finite volume schemes for nearly steady adiabatic flows

We present well-balanced finite volume schemes designed to approximate the Euler equations with gravitation. They are based on a novel local steady state reconstruction. The schemes preserve a discrete equivalent of steady adiabatic flow, which includes non-hydrostatic equilibria. The proposed method works in Cartesian, cylindrical and spherical coordinates. The scheme is not tied to any specific numerical flux and can be combined with any consistent numerical flux for the Euler equations, which provides great flexibility and simplifies the integration into any standard finite volume algorithm. Furthermore, the schemes can cope with general convex equations of state, which is particularly important in astrophysical applications. Both first- and second-order accurate versions of the schemes and their extension to several space dimensions are presented. The superior performance of the well-balanced schemes compared to standard schemes is demonstrated in a variety of numerical experiments. The chosen numerical experiments include simple one-dimensional problems in both Cartesian and spherical geometry, as well as two-dimensional simulations of stellar accretion in cylindrical geometry with a complex multi-physics equation of state

Turbulence: Numerical Analysis, Modelling and Simulation

The problem of accurate and reliable simulation of turbulent flows is a central and intractable challenge that crosses disciplinary boundaries. As the needs for accuracy increase and the applications expand beyond flows where extensive data is available for calibration, the importance of a sound mathematical foundation that addresses the needs of practical computing increases. This Special Issue is directed at this crossroads of rigorous numerical analysis, the physics of turbulence and the practical needs of turbulent flow simulations. It seeks papers providing a broad understanding of the status of the problem considered and open problems that comprise further steps