Vorträge

The lecture will be concerned with the simulation of inviscid and viscous compressible flow in time dependent domains. The motion of the boundary of the domain occupied by the fluid is taken into account with the aid of the ALE (Arbitrary Lagrangian-Eulerian) formulation of the Euler and Navier-Stokes equations describing compressible flow. The system of the governing equations is discretized in space by the discontinous Galerkin method. The time discretization is based on a semi-implicit linearized time stepping scheme, which leads to the solution of a linear algebraic system on each time level. As a result we get an efficient and robust numerical process. The applicability of the developed method will be demonstrated by some computational results obtained for flow in a channel with a moving wall and past an oscillating airfoil.

These results were obtained in cooperation with Vaclav Kucera and Jaroslava Prokopova from Charles University in Prague, Faculty of Mathematics and Physics, and Jaromir Horacek from Institute of Thermomechanics of Academy of Sciences of the Czech Republic.

The Induced Dimension Reduction (IDR) method is a Krylov space method for solving linear systems that was first developed by Sonneveld around 1979 and documented on three and a half pages of a 1980 proceedings paper by Wesseling and Sonneveld. Soon after IDR, Sonneveld introduced his widely applied Conjugate Gradient Squared (CGS) algorithm. Then, in 1990, van der Vorst suggested Bi-CGSTAB that he claimed to improve both those methods.

Bi-CGSTAB has become a method of choice for nonsymmetric linear systems, and it has been generalized in various ways in the hope of further improving its reliability and speed. Among these generalizations there is the ML(k)BiCGSTAB method of Yeung and Chan, which in the framework of block Lanczos methods can be understood as a variation of Bi-CGSTAB with right-hand side block size 1 and left-hand side block size k.

In 2007 Sonneveld and van Gijzen reconsidered IDR and generalized it to IDR(s), claiming that IDR is equally fast but preferable to Bi-CGSTAB, and that IDR(s) may be much faster than IDR = IDR(1). It turned out that IDR(s) is closely related to BiCGSTAB if s = 1 and to ML(s)BiCGSTAB if s > 1. In 2008, a new, particularly ingenious and elegant variant of IDR(s) has been proposed by the same authors.

In this talk we first try to explain the basic, seemingly quite general IDR approach, which differs completely from traditional approaches to Krylov space methods. Then we compare the basic properties of the above mentioned methods and discuss some of their connections.

I will first give a brief description of finite-volume, Godunov-type methods for hyperbolic systems of conservation laws. These methods consist of two types of schemes: upwind and central. My lecture will focus on the second type -- non-oscillatory central schemes.

Godunov-type schemes are projection-evolution methods. In these methods, the solution, at each time step, is interpolated by a (discontinuous) piecewise polynomial interpolant, which is then evolved to the next time level using the integral form of conservation laws. Therefore, in order to design an upwind scheme, (generalized) Riemann problems have to be (approximately) solved at each cell interface. This however may be hard or even impossible.

The main idea in the derivation of central schemes is to avoid solving Riemann problems by averaging over the wave fans generated at cell interfaces. This strategy leads to a family of universal numerical methods that can be applied as a black-box-solver to a wide variety of hyperbolic PDEs and related problems. At the same time, central schemes suffer from (relatively) high numerical viscosity, which can be reduced by incorporating of some upwinding information into the scheme derivation -- this leads to central-upwind schemes, which will be presented in the lecture.

During the talk, I will show a number of recent applications of the central schemes.

Uniqueness for weak solutions of conservation laws is based on the sign of the entropy production across discontinuos solutions. Although the entropy plays a fundamental role in the theory of hyperbolic systems, it is generally not used as a computational tool.
In this talk I describe how the numerical production of entropy induced by the discretization of the equations is a reliable indicator of the quality of the numerical solution. Thus the entropy production can be used as a regularity indicator, identifying the cells in which non linear limiters must be used to prevent the onset of spurious oscillations.
More quantitatively, when the solution is smooth, the entropy production has the same size of the local truncation error and can therefore be used as an a-posteriori error indicator to drive the construction of adaptive grids.