Dietmar Kröner

A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multi dimensions

In this paper we shall derive a posteriori error estimates in the L 1 -norm for upwind finite volume schemes for the discretization of nonlinear conservation laws on unstructured grids in multi dimensions. This result is mainly based on some fundamental a priori error estimates published in a recent paper by C. Chainais-Hillairet. The theoretical results are confirmed by numerical experiments.

An Introduction to Recent Developments in Theory and Numerics for Conservation Laws

An Introduction to Kinetic Schemes for Gas Dynamics.- An Introduction to Nonclassical Shocks of Systems of Conservation Laws.- Viscosity and Relaxation Approximation for Hyperbolic Systems of Conservation Laws.- A Posteriori Error Analysis and Adaptivity for Finite Element Approximations of Hyperbolic Problems.- Numerical Methods for Gasdynamic Systems on Unstructured Meshes.

Convergence of upwind finite volume schemes for scalar conservation laws in two dimensions

This paper proves the convergence of a general class of monotone finite volume methods for numerical schemes of scalar conservation laws in two dimensions on unstructured meshes. There are convergence results for fractional step methods on cartesian grids and for finite element algorithms on unstructured grids. Even for finite volume methods there are some recent results concerning the Lax–Friedrichs and the Godunov finite volume method. The proof in this paper considers a general class including the Lax–Friedrichs and the Engquist–Osher finite volume schemes, and uses a completely different idea than in previous papers to control the entropy dissipation.

Numerical Solutions to Compressible Flows in a Nozzle with Variable Cross-section

Compressible flows in a nozzle can be modeled by the gas dynamics equations in one-dimensional space with source terms. It turns out that along stationary waves, the entropy is conserved. Investigating properties of the system leads us to the determination of stationary waves. Relying on this analysis, we construct a numerical scheme which takes into account the use of stationary waves. Our scheme is shown to be capable of maintaining equilibrium states. This demonstrates the efficiency of the new scheme over classical ones, which usually give unsatisfactory results when reducing the refinement of the mesh-size. Moreover, our scheme converges much faster than the classical ones in most cases.

ABSORBING BOUNDARY CONDITIONS FOR THE LINEARIZED EULER EQUATIONS IN 2-D

In this paper we shall derive some approximate absorbing bound- ary conditions for the initial value problem for the unsteady linearized Euler equations in 2-D. Since we assume that the coefficients of the system are con- stant, we can describe the transformation of the system to a decoupled system of ODEs and the related absorbing boundary conditions explicitly. We shall verify the usefulness of these boundary conditions in some numerical tests for the nonlinear Euler equations in 2-D.

High Performance Computing in Science and Engineering '10

This book presents the state-of-the-art in simulation on supercomputers. Leading researchers present results achieved on systems of the High Performance Computing Center Stuttgart (HLRS) for the year 2010. The reports cover all fields of computational science and engineering, ranging from CFD to computational physics and chemistry to computer science, with a special emphasis on industrially relevant applications. Presenting results for both vector systems and microprocessor-based systems, the book makes it possible to compare the performance levels and usability of various architectures. As HLRS operates the largest NEC SX-8 vector system in the world, this book gives an excellent insight into the potential of vector systems, covering the main methods in high performance computing. Its outstanding results in achieving the highest performance for production codes are of particular interest for both scientists and engineers. The book includes a wealth of color illustrations and tables.

High Performance Computing in Science and Engineering '08

This book presents the state of the art in simulation using supercomputers. Leading researchers present results achieved on systems of the Stuttgart High Performance Computing Center (HLRS) for the year 2008. The reports cover all fields of computational science and engineering, ranging from CFD and computational physics and chemistry to computer science, with a special emphasis on industrially relevant applications. Presenting results for both vector-based and microprocessor-based systems, the book makes it possible to compare the performance levels and usability of various architectures. As the HLRS operates the largest NEC SX-8 vector system in the world, this book gives an excellent insight into the potential of such systems. The book further covers the main methods utilized in high performance computing. Its outstanding results in achieving the highest performance for production codes are of particular interest for both scientists and engineers. The book includes a wealth of coloured illustrations and tables.

A robust numerical method for approximating solutions of a model of two-phase flows and its properties

The objective of the present paper is to extend our earlier works on simpler systems of balance laws in nonconservative form such as the model of fluid flows in a nozzle with variable cross-section to a more complicated system consisting of seven equations which has applications in the modeling of deflagration-to-detonation transition in granular materials. First, we transform the system into an equivalent one which can be regarded as a composition of three subsystems. Then, depending on the characterization of each subsystem, we propose a convenient numerical treatment of the subsystem separately. Precisely, in the first subsystem of the governing equations in the gas phase, stationary waves are used to absorb the nonconservative terms into an underlying numerical scheme. In the second subsystem of conservation laws of the mixture we can take a suitable scheme for conservation laws. For the third subsystem of the compaction dynamics equation, the fact that the velocities remain constant across solid contacts suggests us to employ the technique of Engquist-Oshers scheme. Then, we prove that our method possesses some interesting properties: it preserves the positivity of the volume fractions in both phases, and in the gas phase, our scheme is capable of capturing equilibrium states, preserves the positivity of the density, and satisfies the numerical minimum entropy principle. Numerical tests show that our scheme can provide reasonable approximations for data the supersonic regions, but the results are not satisfactory in the subsonic region. However, the scheme is numerically stable and robust.

Convergence of a staggered Lax-Friedrichs scheme for nonlinear conservation laws on unstructured two-dimensional grids

Summary. Based on Nessyahu and Tadmors nonoscillatory central difference schemes for one-dimensional hyperbolic conservation laws [16], for higher dimensions several finite volume extensions and numerical results on structured and unstructured grids have been presented. The experiments show the wide applicability of these multidimensional schemes. The theoretical arguments which support this are some maximum-principles and a convergence proof in the scalar linear case. A general proof of convergence, as obtained for the original one-dimensional NT-schemes, does not exist for any of the extensions to multidimensional nonlinear problems. For the finite volume extension on two-dimensional unstructured grids introduced by Arminjon and Viallon [3,4] we present a proof of convergence for the first order scheme in case of a nonlinear scalar hyperbolic conservation law.

Numerical solution of Navier-Stokes-Korteweg systems by Local Discontinuous Galerkin methods in multiple space dimensions

Compressible liquid-vapor flow with phase transitions can be described by systems of Navier-Stokes-Korteweg type. They extend the Navier-Stokes equations by nonlinear higher-grade terms which take the form of either differential or nonlocal integral operators. A numerical approximation method on the basis of the Local Discontinuous Galerkin method in multiple space dimensions is suggested for isothermal flows. It relies on a specific discretization of a non-conservative formulation. To enhance the performance of the overall scheme two techniques are used: (i) local spatial adaptivity based on gradient indicators for the density and (ii) parallelism based on domain decomposition.The paper concludes with numerical experiments in two and three space dimensions. They show the reliability and efficiency of the proposed approach as well as they demonstrate the applicability of the models for several important phase transition phenomena.