Christian Rohde

Fully Discrete, Entropy Conservative Schemes of Arbitrary Order

We consider weak solutions of (hyperbolic or hyperbolic-elliptic) systems of conservation laws in one-space dimension and their approximation by finite difference schemes in conservative form. The systems under consideration are endowed with an entropy-entropy flux pair. We introduce a general approach to construct second and third order accurate, fully discrete (in both space and time) entropy conservative schemes. In general, these schemes are fully nonlinear implicit, but in some important cases can be explicit or linear implicit. Furthermore, semidiscrete entropy conservative schemes of arbitrary order are presented. The entropy conservative schemes are used to construct a numerical method for the computation of weak solutions containing nonclassical regularization-sensitive shock waves. Finally, specific examples are investigated and tested numerically. Our approach extends the results and techniques by Tadmor [in Numerical Methods for Compressible Flows---Finite Difference, Element and Volume Techniques, ASME, New York, 1986, pp. 149--158], LeFloch and Rohde [SIAM J. Numer. Anal., 37 (2000), pp. 2023--2060].

High-Order Schemes, Entropy Inequalities, and Nonclassical Shocks

We are concerned with the approximation of undercompressive, regularization-sensitive, nonclassical solutions of hyperbolic systems of conservation laws by high-order accurate, conservative, and semidiscrete finite difference schemes. Nonclassical shock waves can be generated by diffusive and dispersive terms kept in balance. Particular attention is given here to a class of systems of conservation laws including the scalar equations and the system of nonlinear elasticity and to linear diffusion and dispersion in either the conservative or the entropy variables. First, we investigate the existence and the properties of entropy conservative schemes---a notion due to Tadmor [ Math. Comp., 49 (1987), pp. 91--103]. In particular we exhibit a new five-point scheme which is third-order accurate, at least. Second, we study a class of entropy stable and high-order accurate schemes satisfying a single cell entropy inequality. They are built from any high-order entropy conservative scheme by adding to it a mesh-independent, numerical viscosity, which preserves the order of accuracy of the base scheme. These schemes can only converge to solutions of the system of conservation laws satisfying the entropy inequality. These entropy stable schemes exhibit mild oscillations near shocks and, interestingly, may converge to classical or nonclassical entropy solutions, depending on the sign of their dispersion coefficient. Then, based on a third-order, entropy conservative scheme, we propose a general scheme for the numerical computation of nonclassical shocks. Importantly, our scheme satisfies a cell entropy inequality. Following Hayes and LeFloch [SIAM J. Numer. Anal., 35 (1998), pp. 2169--2194], we determine numerically the kinetic function which uniquely characterizes the dynamics of nonclassical shocks for each regularization of the conservation laws. Our results compare favorably with previous analytical and numerical results. Finally, we prove that there exists no fully discrete and entropy conservative scheme and we investigate the entropy stability of a class of fully discrete, Lax--Wendroff type schemes.

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.

Overlapping Schwarz Waveform Relaxation for Convection-Dominated Nonlinear Conservation Laws

We analyze the convergence of the overlapping Schwarz waveform relaxation algorithm applied to convection-dominated nonlinear conservation laws in one spatial dimension. For two subdomains and bounded time intervals we prove superlinear asymptotic convergence of the algorithm in the parabolic case and convergence in a finite number of steps in the hyperbolic limit. The convergence results depend on the overlap, the viscosity, and the length of the time interval under consideration, but they are independent of the number of subdomains, as a generalization of the results to many subdomains shows. To investigate the behavior of the algorithm for a long time, we apply it to the Burgers equation and use a steady state argument to prove that the algorithm converges linearly over long time intervals. This result reveals an interesting paradox: while for the superlinear convergence rate on bounded time intervals decreasing the viscosity improves the performance, in the linear convergence regime decreasing the viscosity slows down the convergence rate and the algorithm can converge arbitrarily slowly, if there is a standing shock wave in the overlap. We illustrate our theoretical results with numerical experiments.

Fast Relaxation Solvers for Hyperbolic-Elliptic Phase Transition Problems

Phase transition problems in compressible media can be modelled by mixed hyperbolic-elliptic systems of conservation laws. Within this approach phase boundaries are understood as shock waves that satisfy additional constraints, sometimes called kinetic relations. For numerical approximation tracking-type algorithms have been suggested. The core piece of these algorithms is the usage of exact Riemann solvers incorporating the kinetic relation. However, exact Riemann solvers are computationally expensive or even not available. In this paper we present a class of approximate Riemann solvers for hyperbolic-elliptic models that relies on a generalized relaxation procedure. It preserves in particular the kinetic relation for phase boundaries exactly and gives for isolated phase transitions the correct solutions. In combination with a novel subiteration procedure the approximate Riemann solvers are used in the tracking algorithms. The efficiency of the approach is validated on a barotropic system with linear kin...

Sharp and diffuse interface methods for phase transition problems in liquid-vapour flows

We consider the dynamics of a compressible fluid exhibiting phase transitions between a liquid and a vapour phase. As the basic mathematical model we use the Euler equations for a sharp interface approach and local and global versions of the NavierStokes-Korteweg equations for the diffuse interface approach. The mathematical models are discussed and we introduce discretization methods for both approaches. Finally numerical simulations in one and two space dimensions are presented.This work is motivated by the numerical simulation of the generation and break-up of droplets after the impact of a rigid body on a tank filled with a compressible fluid. This paper splits into two very different parts. The first part deals with the modeling and the numerical resolution of a spray of liquid droplets in a compressible medium like air. Phenomena taken into account are the breakup effects due to the velocity and pressure waves in the compressible ambient fluid. The second part is concerned with the transport of a rigid body in a compressible liquid, involving reciprocal effects between the two components. A new one-dimensional algorithm working on a fixed Eulerian mesh is proposed. The GENJET (GENeration and breakup of liquid JETs) project has been proposed by the Centre d’Études de Gramat (CEG) of the Délégation Générale de l’Armement (DGA). It concerns the general study of the consequences of a violent impact of a rigid body against a reservoir of fluid. Experiments show that once the solid has pierced the shell of the reservoir, it provokes a dramatic increase of the pressure inside the reservoir, whose effect is the ejection of some fluid through the pierced hole. The generated liquid jet then expands into the ambient air, where it can interact with some air pressure waves, leading to a fragmentation of the jet into small droplets. These experiments show that after having pierced the shell, the projectile behaves as a rigid body. They also show that the liquid inside the reservoir behaves as a compressible fluid (indeed, the projectile velocity, around 1000 m.s, is in the same order of magnitude than the sound speed in the liquid). The modeling of such a complex flow requires to take into account very different regimes, from the pure compressible and/or incompressible flow condition to a droplet regime (such a regime sharing some similarities with kinetic modeling of Liquid jet generation and breakup 3 particles). Moreover many scales are needed to correctly describe the complete experiments, from the large hydrodynamic scale to the small droplet scale. The study done during CEMRACS 2004 focused on the fluid regime and on the droplet regime, since some important difficulties are still there for both regimes separately. • Concerning the breakup of droplets in the air, we have focused on physical and numerical modeling issues. • Concerning the fluid regime, an important difficulty at the numerical level is that we want to get an accurate numerical description of the transport of a rigid body inside a compressible fluid. Even if the rigid body is of course not a fluid, the situation shares at the numerical level a lot of similarities with the coupling an incompressible fluid with a compressible one. Thus this part of the study concerns more numerical algorithms than the modeling. The present paper follows this cutout of the study. Section 1 presents the modeling of the breakup of droplets, whereas section 2 treats the coupling of the rigid body and the fluid. In both sections, numerical results are reported. In view of the main goal of the GENJET project, a natural perspective of the work described below would be the coupling of the models, algorithms and numerical methods. 1. A kinetic modeling of a breaking up spray with high Weber numbers In this section, we aim to model a spray of droplets which evolve in an ambient fluid (typically the air). That kind of problem was first studied by Williams for combustion issues [32]. The works of O’Rourke [20] helped to set the modeling of such situations and their numerical simulation through an industrial code, KIVA [1]. The main phenomenon that occuring in the spray is the breakup of the droplets. Any other phenomena, such as collisions or coalescence, will be neglected in this work, but they are reviewed in [3] for example. Instead of using the TAB model (see [2]), which is more accurate for droplets with low Weber numbers, we choose the so-called Reitz wave model [27], [21], [4]. Then this breakup model is taken into account in a kinetic model [14], [2]. The question of the spray behavior with respect to the breaking up has arised in the context of the French military industry. One aims to model with an accurate precision the evolution of a spray of liquid droplets inside the air. In that situation, the droplets of the spray are assumed to remain incompressible (the mass density ρd is a constant of the problem) and spherical. We also assume that the forces on the spray are negligible with respect to the drag force, at least at the beginning of the computations. After a few seconds, the gravitation may become preponderant. Note that the aspects of energy transfer will not be tackled in this report.In this paper we apply the ADER one step time discretization to the Discontinuous Galerkin framework for hyperbolic conservation laws. In the case of linear hyperbolic systems we obtain a quadrature-free explicit single-step scheme of arbitrary order of accuracy in space and time on Cartesian and triangular meshes. The ADERDG scheme does not need more memory than a first order explicit Euler time-stepping scheme. This becomes possible because of an extensive use of the governing equations inside the numerical scheme. In the nonlinear case, quadrature of the ADER-DG scheme in space and time is performed with Gaussian quadrature formulae of suitable order of accuracy. We show numerical convergence results for the linearized Euler equations up to 10th order of accuracy in space and time on Cartesian and triangular meshes. Numerical results for the nonlinear Euler equations up to 6th order of accuracy in space and time are provided as well. In this paper we also show the possibility of applying a linear reconstruction operator of the order 3N +2 to the degrees of freedom of the DG method resulting in a numerical scheme of the order 3N + 3 on Cartesian grids where N is the order of the original basis functions before reconstruction.In this paper, we introduce a new PIC method based on an adaptive multi-resolution scheme for solving the one dimensional Vlasov–Poisson equation. Our approach is based on a description of the solution by particles of unit weight and on a reconstruction of the density at each time step of the numerical scheme by an adaptive wavelet technique: the density is firstly estimated in a proper wavelet basis as a distribution function from the current empirical data and then “de-noised” by a thresholding procedure. The so-called Landau damping problem is considered for validating our method. The numerical results agree with those obtained by the classical PIC scheme, suggesting that this multi-resolution procedure could be extended with success to plasma dynamics in higher dimensions.

Numerical approximation of entropy solutions for hyperbolic integro-differential equations

Summary.Scalar hyperbolic integro-differential equations arise as models for e.g. radiating or self-gravitating fluid flow. We present finite volume schemes on unstructured grids applied to the Cauchy problem for such equations. For a rather general class of integral operators we show convergence of the approximate solutions to a possibly discontinuous entropy solution of the problem. For a specific model problem in radiative hydrodynamics we introduce a convergent fully discrete finite volume scheme. Under the assumption of sufficiently fast spatial decay of the entropy solution we can even establish the convergence rate h1/4|ln(h)| where h denotes the grid parameter. The convergence proofs rely on appropriate variants of the classical Kruzhkov method for local balance laws together with a truncation technique to cope with the nonlocal character of the integral operator.

Scalar Conservation Laws with Mixed Local and Nonlocal Diffusion-Dispersion Terms

We consider a nonlinear scalar conservation law that is regularized by a local viscous term and a nonlocal dispersive term. This nonstandard regularization is motivated by phase transition problems that take into account long range interactions close to the interface. We identify a parameter regime such that this mixed-type regularization provides a new example that is able to drive nonclassical undercompressive shock waves in the limit of vanishing regularization parameter. In view of the applications this shows that nonlocal regularizations can be used to model dynamical phase transition processes.In the next step we establish the existence and uniqueness of classical solutions for the Cauchy problem in multiple space dimensions. In the main part of the paper we then deduce appropriate a priori estimates to analyze the sharp-interface limit for vanishing regularization parameter with the method of compensated compactness in one space dimension and, using measure-valued solutions, in multiple space dimen...

Multirate time integration for coupled saturated/unsaturated porous medium and free flow systems

A model formulation to describe fluid flows in coupled saturated/unsaturated porous medium and adjacent free flow regions is proposed. The Stokes equations are applied in the free flow domain, while the Richards equation is used to model the porous medium system. These two flow problems are coupled at the fluid-porous interface via an appropriate set of interface conditions. A multiple-time-step scheme is developed to solve the coupled problem efficiently. Numerical simulation results are presented for a model problem and a realistic setting that demonstrate the convergence and efficiency of the proposed computational algorithm. Time-splitting multistep methods can be successfully applied for modeling other physical systems where the processes evolve on different time scales, and these potential extensions are discussed.

Error Estimates for Finite Volume Approximations of Classical Solutions for Nonlinear Systems of Hyperbolic Balance Laws

We consider a general class of finite volume schemes on unstructured but quasi-uniform meshes for first-order systems of hyperbolic balance laws on unstructured meshes. Provided the system is equipped with at least one entropy-entropy flux tuple and the associated Cauchy problem allows for a classical solution