Analysis and Numerics for Conservation Laws

We investigate the interaction of shock waves in a heavy gas with embedded light gas bubbles next to a rigid wall. This may give insight regarding cavitation processes in water. Due to the highly dynamical, unsteady processes under consideration we use an adaptive FV scheme for the computations to resolve accurately all physically relevant effects. The results are validated by comparison with tube experiments.

A finite volume method has been developed in this work for solving the conservation equations of argon plasma flows in magnetoplasmadynamic self-field accelerators. These accelerators can be used for interplanetary spaceflight missions because of their high specific impulse and high thrust density.

Calculations show in agreement with the experiment that a primary reason for plasma instabilities at high current settings — which are limiting the operational envelope and the thruster lifetime — is the depletion of density and charge carriers in front of the anode because of the pinch effect.

The calculated thrust data agree well with experimental values, so that the newly developed method can be used for the design and optimization of new thrusters.

The paper deals with the transition regime of gas flows between the mesoscopic and the macroscopic levels. We survey theoretical results and provide numerical tools. As the basic numerical scheme for the solution of the Boltzmann equation we use a hexagonal model proposed in [1].

Numerical simulations can be the key to the thorough understanding of the multi-dimensional nature of transient detonation waves. But the accurate approximation of realistic detonations is extremely demanding, because a wide range of different scales need to be resolved. This paper describes an entire solution strategy for the Euler equations of thermally perfect gas-mixtures with detailed chemical kinetics that is based on a highly adaptive finite volume method for blockstructured Cartesian meshes. Large-scale simulations of unstable detonation structures of hydrogen-oxygen detonations demonstrate the efficiency of the approach in practice.

The Rayleigh-Bénard problem and the Taylor-Couette problem are two well-known stability problems that are traditionally treated with linear stability analysis. In the vast majority of these stability calculations the fluid is considered to be incompressible [Cha61, DR81]. Only with this assumption and simplification is possible to conduct a linear stability analysis analytically.

In order to calculate the stability limits of a compressible fluid by use of a linear stability analysis therefore in this work a numerical linear stability analysis is presented. The numerical stability analysis is based upon the equations of balance for mass, momentum and energy that are completed with the constitutive equations by Navier-Stokes and Fourier. The algorithm allows to calculate the regions of stability for arbitrary one-dimensional and stationary basic states.

This numerical stability analysis is used to calculate the stability region for the Rayleigh-Bénard problem. The main result is that the critical Rayleigh number does not have a constant value, as calculations involving the Boussinesq approximation suggest misleadingly, but that the value of the critical Rayleigh number depends strongly on the thickness of the fluid layer. Furthermore, an empirically found relationship between the critical Rayleigh number and the thickness of the fluid layer is presented (14). Its efficiency is successfully verified with the results of the numerical linear stability analysis. The results for the critical Rayleigh number show clearly that the compressibility of a fluid must not be neglected in the stability analysis of the Rayleigh-Bénard problem.

Secondly, the more complicated Taylor-Couette problem is treated with the numerical linear stability analysis. In contrast to the traditional stability analysis by Taylor [Tay23], the fluid is considered to be compressible and includes the temperature as a field variable. The effectiveness of the numerical linear stability analysis is manifested by the good agreement of the comparison with experimental results. In addition to that, temperature effects are studied and are compared with experiments.

An exact Riemann solver is developed for the investigation of non-classical wave phenomena in BZT fluids and fluids which undergo a phase transition. Here we outline the basic construction principles of this Riemann solver employing a general equation of state that takes negative nonlinearity and phase transition into account. This exact Riemann solver is a useful validation tool for numerical schemes, in particular, when applied to the aforementioned fluids. As an application, we present some numerical results where we consider flow fields exhibiting non-classical wave phenomena due to BZT fluids and phase transition.

The equations of compressible radiation magnetohydrodynamics provide a widely accepted mathematical model for the basic fluid-dynamical processes in the sun's atmosphere. From the mathematical point of view the equations constitute an instance of a system of non-local hyperbolic balance laws. We have developed and implemented numerical methods in three space dimensions on the basis of a finite volume scheme that allow the efficient approximation of weak solutions. Key features are the use of efficient Riemann solvers, a special treatment of the divergence constraint, higher-order schemes, the extended short characteristics method, local mesh adaption, and parallelization using dynamic load balancing. Moreover, methods to cope with the special nature of the atmosphere are included.

In this contribution we give an overview of our work, highlight our most important results, and report on some new developments. In particular, we present a scalar model problem for which an almost complete analytical treatment is possible.

The hyperbolic system that describes heat conduction at low temperatures and the relativistic Euler equations belong to a class of hyperbolic conservation laws that result from an underlying kinetic equation. The focus of this study is the establishment of an kinetic approach in order to solve initial and boundary value problems for the two examples. The ingredients of the kinetic approach are: (i) Representation of macroscopic fields by moment integrals of the kinetic phase density. (ii) Decomposition of the evolution into periods of free flight, which are interrupted by update times. (iii) At the update times the data are refreshed by the Maximum Entropy Principle.

This paper proposes a general multi-dimensional front tracking concept for various physical problems involving specially discontinuous solution features. The tracking method is based on the level-set approach with a restricted dynamic definition range in the vicinity of the fronts on fixed grids of arbitrary cell structure.

To combine the front tracking procedure with the continuous part of the field to be simulated, a double sided flux discretization called flux-separation and a set of inner boundary conditions over the discontinuities are used. The methods developed are not restricted to fluid dynamics, however all examples relate to this class of simulation problems.

Special attention is drawn to the restrictions of the classical level-set method, i.e. accuracy issues and topological restrictions. In this concern, an improved time integration method for the front motion is introduced and the problem of interacting discontinuities is addressed. The methods are integrated in the object oriented Finite-Volume solution package MOUSE [1] for systems of conservation laws on arbitrary grids.

The standard models for groups of interacting and moving individuals (from cell biology to vertebrate population dynamics) are reaction-diffusion models. They base on Brownian motion, which is characterized by one single parameter (diffusion coefficient). In particular for moving bacteria and (slime mold) amoebae, detailed information on individual movement behavior is available (speed, run times, turn angle distributions). If such information is entered into models for populations, then reaction-transport equations or hyperbolic equations (telegraph equations, damped wave equations) result.

The goal of this review is to present some basic applications of transport equations and hyperbolic systems and to illustrate the connections between transport equations, hyperbolic models, and reaction-diffusion equations. Applied to chemosensitive movement (chemotaxis) functional estimates for the nonlinearities in the classical chemotaxis model (Patlak-Keller-Segel) can be derived, based on the individual behavior of cells and attractants.

A detailed review is given on two methods of reduction for transport equations. First the construction of parabolic limits (diffusion limits) for linear and non-linear transport equations and then a moment closure method based on energy minimization principles. We illustrate the moment closure method on the lowest non-trivial case (two-moment closure), which leads to Cattaneo systems.

Moreover we study coupled dynamical systems and models with quiescent states. These occur naturally if it is assumed that different processes, like movement and reproduction, do not occur simultaneously. We report on travelling front problems, stability, epidemic modeling, and transport equations with resting phases.

We discuss several results on the existence of continuous travelling wave solutions in systems of conservation laws with nonlinear source terms.

In the first part we show how waves with oscillatory tails can emerge from the combination of a strictly hyperbolic system of conservation laws and a source term possessing a stable line of equilibria. Two-dimensional manifolds of equilibria can lead to Takens-Bogdanov bifurcations without parameters. In this case there exist several families of small heteroclinic waves connecting different parts of the equilibrium manifold.

Our viewpoint is from dynamical systems and bifurcation theory. Local normal forms at singularities are used and the dynamics is described with the help of blowup transformations and invariant manifolds.

Substellar atmospheres are observed to be irregularly variable for which the formation of dust clouds is the most promising candidate explanation. The atmospheric gas is convectively unstable and, last but not least, colliding convective cells are seen as cause for a turbulent fluid field. Since dust formation depends on the local properties of the fluid, turbulence influences the dust formation process and may even allow the dust formation in an initially dust-hostile gas.

A regime-wise investigation of dust forming substellar atmospheric situations reveals that the largest scales are determined by the interplay between gravitational settling and convective replenishment which results in a dust-stratified atmosphere. The regime of small scales is determined by the interaction of turbulent fluctuations. Resulting lane-like and curled dust distributions combine to larger and larger structures. We compile necessary criteria for a subgrid model in the frame of large scale simulations as result of our study on small scale turbulence in dust forming gases.

In this article, two meshfree methods for the numerical solution of conservation laws are considered. The Finite Volume Particle Method (FVPM) generalizes the Finite Volume approach and the Finite Pointset Method (FPM) is a Finite Difference scheme which can work on unstructured and moving point clouds. Details of the derivation and numerical examples are presented for the case of incompressible, viscous, two-phase flow. In the case of FVPM, our main focus lies on the derivation of stability estimates.

Type Ia supernovae, i.e. stellar explosions which do not have hydrogen in their spectra, but intermediate-mass elements such as silicon, calcium, cobalt, and iron, have recently received considerable attention because it appears that they can be used as ”standard candles” to measure cosmic distances out to billions of light years away from us. Observations of type Ia supernovae seem to indicate that we are living in a universe that started to accelerate its expansion when it was about half its present age. These conclusions rest primarily on phenomenological models which, however, lack proper theoretical understanding, mainly because the explosion process, initiated by thermonuclear fusion of carbon and oxygen into heavier elements, is difficult to simulate even on supercomputers.

Here, we investigate a new way of modeling turbulent thermonuclear deflagration fronts in white dwarfs undergoing a type Ia supernova explosion. Our approach is based on a level set method which treats the front as a mathematical discontinuity and allows for full coupling between the front geometry and the flow field. New results of the method applied to the problem of type Ia supernovae are obtained. It is shown that in 2-D with high spatial resolution and a physically motivated subgrid scale model for the nuclear flames numerically “converged” results can be obtained, but for most initial conditions the stars do not explode. In contrast, simulations in 3-D do give the desired explosions and many of their properties, such as the explosion energies, lightcurves and nucleosynthesis products, are in very good agreement with observed type Ia supernovae.

The charge conservation laws in general are not strictly obeyed in computational electromagnetics and Magnetohydrodynamics (MHD), due to the presence of various types of numerical errors. In this paper, a field theoretical method for the treatment of the often violated charge conservation laws in computational electrodynamics and MHD has been investigated, which reduces to the well-known hyperbolic Generalized Lagrange Multplier (GLM) scheme under particular constraints. The central idea of our divergence correction scheme is the implementation of the physically consistent counter terms to Maxwell and MHD equations, for the restoration of the charge conservation laws. The underlying idea has been verified by numerical experiments for Maxwell-Vlasov and shallow water MHD systems.

A numerical technique for the simulation of accelerating turbulent premixed flames in large scale geometries is presented. It is based on a hybrid capturing/tracking method. It resembles a tracking scheme in that the front geometry is explicitly computed and propagated using a level set method. The basic flow properties are provided by solving the reactive Euler equations. The flame-flow-coupling is achieved by an in-cell-reconstruction technique, i.e., in cells cut by the flame the discontinuous solution is reconstructed from given cell-averages by applying Rankine-Hugoniot type jump conditions. Then the reconstructed states and again the front geometry are used to define accurate effective numerical fluxes across grid cell interfaces intersected by the front during the time step considered. Hence the scheme also resembles a capturing scheme in that only cell averages of conserved quantities are updated. To enable the modelling of inherently unsteady effects, like quenching, reignition, etc., during flame acceleration, the new key idea is to provide a local, quasi-onedimensional flame structure model and to extend the Rankine-Hugoniot conditions so as to allow for inherently unsteady flame structure evolution. A source term appearing in the modified jump conditions is computed by evaluating a suitable functional on the basis of a onedimensional flame structure module, that is attached in normal direction to the flame front. This module additionally yields quantities like the net mass burning rate, necessary for the propagation of the level set, and the specific heat release important for the energy release due to the consumption of fuel. Generally the local flame structure calculation takes into account internal (small scale) physical effects which are not active in the (large scale) outer flow but essential for the front motion and its feedback on the surrounding fluid. If a suitable set of different (turbulent) combustion models to compute the flame structure is provided, the new numerical technique allows us to consistently represent laminar deflagrations, fast turbulent deflagrations as well as detonation waves. Supplemented with suitable criteria that capture the essence of a Deflagration-to-Detonation-Transition (DDT), the complete evolution of such an event can be implemented in principle.

In this article, we use the recently developed framework of state and flux decompositions to point out some interesting connections and differences between several Riemann-solver free numerical approaches for systems of hyperbolic conservation laws. We include a numerical comparison of Fey's Method of Transport with a second order version of the HLL scheme and prove an interesting property of the former scheme for linear waves contained in the equations of ideal gas dynamics.

Relaxation dynamics, scaling limits, and relaxation schemes are three main topics on hyperbolic relaxation problems that, remarkably, can be well understood with one model equation. The criterion that leads to desired results for the three problems is the so called “sub-characteristic condition”. The criterion of this nature is also pivotal in the study of general hyperbolic relaxation problems.

In this article we review the recent research development in hyperbolic relaxation problems. The emphasis is on contributions associated with our own project within ANumE priority research program. We will first review some basic properties and notions for hyperbolic relaxation problems, and then focus our investigation on three main topics associated with the underlying relaxation model: relaxation dynamics, scaling limits as well as convergence theory of relaxation schemes.

A variety of numerical schemes operates on staggered grids, that is a pair of meshes of the same computational domain whose interior nodes (in 1D), edges (in 2D) and faces (in 3D) do not coincide. While the shape of a staggered grid is canonical on an uniformly refined Cartesian mesh it becomes more complicated for an underlying adaptively refined grid, in particular in higher spatial dimensions. Here we present both a construction technique for staggered dual grids exploiting the structure of the adaptively refined Cartesian primal grids in 2D and 3D, and discuss the necessary modifications of a standard Finite Volume scheme which is originally formulated on uniform meshes.

This report summarizes our works on hyperbolic systems of first-order partial differential equations with source terms. We discuss the introduction of our structural stability and entropy dissipation conditions for initial or initial-boundary value problems. For initial value problems, several systematic results are reviewed. These include the non-existence of (linearly stable) relaxation approximations to non-strongly hyperbolic systems of equations, the justification of the formal zero relaxation limit, the existence of relaxation shock profiles, and the existence of global smooth solutions for balance laws.