Nonlinear Hyperbolic Problems a perspective view on conservation laws

Transcription

1 Short courses (pp. 2 3) INdAM International Workshop on Nonlinear Hyperbolic Problems a perspective view on conservation laws Stefano Bianchini, Riemann problem for hyperbolic system of conservation laws with boundary Bjorn Engquist, High resolution shock capturing methods Guy Metivier, Stability of shock waves Plenary conferences (pp. 4 8) Yann Brenier, A Hilbertian approach to hyperbolic non linear conservation laws Constantine Dafermos, Hyperbolic balance laws with dissipation Rosa Donat, A penalization technique for compressible fluid flow Helge Holden, Conservative and dissipative solutions of the Camassa Holm equation Tai Ping Liu, Generalized Riemann Problem Benedetto Piccoli, Traffic flow on networks Denis Serre, Shock reflection for a Chaplygin gas Angela Stevens, Orientational aggregation and pattern formation due to local cell interaction Athanasios E. Tzavaras, Transport equations and oscillations in hyperbolic equations and systems of two conservation laws Kevin Zumbrun, New vistas in stability of viscous shock and detonation waves Invited talks (pp. 9 11) Gilles Carbou, Relaxation approximation of the Kerr model for the impedance initial boundary value problem Volker Elling, Regular reflection and the sonic criterion Ingenuin Gasser, Modeling, analysis and simulations of a tunnel fire model Helge Kristian Jenssen, Systems of 1-D conservation laws with geometric constraints Andreas Münch, Shocks in thin films Michael Westdickenberg, Finite energy solutions to the isentropic Euler equations with geometric effects

2 Short courses Riemann problem for hyperbolic system of conservation laws with boundary Stefano Bianchini I.S.A.S., Trieste, Italy This short course will introduce the construction of the solution to u t + f(u) x = 0, u(0, x) = u 0, u(t, 0) = u b This solution is generated by taking the limit of the hyperbolic-parabolic system u t + f(u) x = ɛ(b(u)u x ) x as ɛ 0. It requires techniques from singular ODE and perturbation of linear operators. 1) Introduction to the Riemann problem for hyperbolic systems: basic properties and construction for special cases. Construction in the presence of boundary. Some examples. 2) Analysis of singular ODE and contruction of invariant manifolds. 3) Perturbation of linear operators and eigenvalue analysis of hyperbolic-parabolic systems. 4) Contruction of the solution to the boundary Riemann problem. Two examples from physics and a (more difficult) artificial example. High resolution shock capturing methods Bjorn Engquist K.T.H., N.A.Da., Stockholm, Sweden engquist@nada.kth.se The lectures will start with a survey of different numerical methods for nonlinear hyperbolic conservation laws and the properties that these methods must satisfy. The basic convergence theory will then be presented. This will include compactness based arguments and analysis using implicit function theorems. Gudonov, TVD, ENO, WENO an DG methods will be discussed. We will also consider the analogous algorithms for Hamilton Jacobi equations. Applications to fluid mechanics will be the focus but other applications will also be considered. 2

3 Stability of shock waves Guy Metivier University of Bordeaux 1, France The goal of these lectures is to discuss the stability of curved multidimensional shock waves for hyperbolic systems of conservation laws as zero viscosity limits of solutions of viscous equations, e.g. shock waves for compressible Euler equations as limits of solutions of Navier Stokes equations. The analysis involves several steps: planar shocks as limits of traveling viscous waves; the stability analysis of these objects, seen as a spectral (or Fourier Laplace) stability analysis; the linear stability, that is the well posed-ness of the linearized equations; and finally the nonlinear stability analysis, that is the well posed-ness of the perturbed nonlinear equations. After presenting these notions, the emphasis will be put on the spectral stability analysis. In particular we will show that the apparent singularity at the origin of the resolvant equation for the viscous equation can be resolved by introducing a viscous artificial front; next this augmented system can be split into two parts, the inviscid problem including the shock front perturbation being a nonsingular limit of one of these two parts. 3

4 Plenary conferences A Hilbertian approach to hyperbolic non linear conservation laws Yann Brenier C.N.R.S., Université de Nice, France brenier@math.unice.fr It is customary to adress hyperbolic conservation laws (or Hamilton Jacobi equations) in functional spaces that are neither hilbertian nor reflexive (typically L 1, BV, C 0, Lip, etc... ). We show that, in some simple but significative cases (multidimensional scalar conservation laws, Chaplygin gas or Born Infeld electromagnetism in one space variable), a simple L 2 formulation can be introduced, leading to straightforward well posedness and stability results. This approach can be extended to some coupled system like pressureless Euler Poisson systems. In each case, very accurate numerical schemes can be designed according to the L 2 formulation. Reference: Hyperbolic balance laws with dissipation Constantine Dafermos Brown University, U.S.A. dafermos@dam.brown.edu Global BV solutions will be constructed to the Cauchy problem for hyperbolic systems of balance laws with dissipative source terms induced by relaxation mechanisms. A penalization technique for compressible fluid flow Rosa Donat University of Valencia, Spain donat@uv.es The presence of obstacles of arbitrary shapes in moderate to high Reynolds number flows is an important problem in numerical simulations of convection-dominated problems. Numerically imposing the appropriate boundary conditions on the boundary of the obstacles is a complex issue that has been handled by a variety of techniques, from coordinate transformations and body fitted grids to embedded boundary approaches in Cartesian meshes. Penalization methods represent a different line of work, developed by various authors in order to be able to avoid the use of boundary fitted grids, and other non-standard 4

5 adjustments when handling obstacles in incompressible fluid flow. The physical idea of the volume penalization technique introduced by E. Arquis and J.P. Caltagirone [2] is to model the obstacle as a porous medium with porosity tending to zero. For incompressible flows, it leads to a Brinkman type model with variable permeability, where the fluid domain has a large permeability in front of the one of the porous medium.the penalized system is solved in an obstacle-free domain, hence fast and effective methods for Cartesian grids can be used. In [3], we propose a Brinkman type penalization method for the numerical simulation Navier Stokes flows at high Reynolds numbers. Following the guidelines stated in [1] for the the incompressible case, a penalization term is added to the momentum equations, with the aim of enforcing no-slip boundary conditions on the boundary of the obstacle, and to the energy equation, in order to enforce Dirichlet boundary conditions on the temperature. The penalized system can then be solved in the whole computational domain by using the High Resolution Shock Capturing technology on the convective terms, which is particularly efficient and easy to implement on uniform Cartesian Meshes. We will show a series of numerical experiments that show that the no slip boundary conditions on the velocity, and the Dirichlet condition on the temperature are correctly recovered on the boundary of the obstacle. The penalization technique we propose leads, hence, to a fast an reliable numerical technique for handling compressible fluid flow in the presence of obstacles. References [1] P. Angot, C.-H. Bruneau, P. Fabrie: A penalization method to take into account obstacles in incompressible flows. Numer. Math., 81, (1999) [2] E. Arquis, J.P.Caltagirone: Sur les conditions hydrodynamiques au voisinage d une interface milieu fluide-milieu poreux: application à la convection naturelle. C.R. Acad. Sci. Paris II, 299, 1 4 (1984) [3] G. Chiavassa, R. Donat: A penalization Method for Compressible Fluid Flow. Proceedings of Fifth Intl. Conf. on Engineering Computational Technology ECT2006, B.H.V Topping, G. Montero and R. Montenegro, Civil-Comp Press, 2006 Conservative and dissipative solutions of the Camassa Holm equation Helge Holden University of Oslo, Norway holden@math.ntnu.no We discuss the Cauchy problem for the Camassa-Holm equation u t u xxt + 3uu x 2u x u xx uu xxx = 0. Weak solutions are in general not unique, and we study conservative solutions where the energy is preserved as well as dissipative solutions. The approach is based on the interplay between the Eulerian and Lagrangian formulation of the equation. The work is joint with Xavier Raynaud (Trondheim). 5

6 Generalized Riemann Problem Tai Ping Liu Stanford University, U.S.A. and Academia Sinica, China One of the main goals of the study of the shock wave theory is to identify stable and unstable stationary gas flows around a solid. Even when the construction of the stationary flow is available, the stability issue is often difficult. One way to settle the stability issue, at least from the physical point of view, is to consider generalized Riemann problem. A generalized Riemann solution consists of stationary and self-similar solutions so that the whole wave pattern is non-interacting. We will give examples in relaxation models, conservation laws with moving source, and multi-dimensional gas flows with shock reflections. Traffic flow on networks Benedetto Piccoli I.A.C. Mauro Picone C.N.R., Rome, Italy b.piccoli@iac.rm.cnr.it A mathematical model for fluid dynamic flows on networks which is based on conservation laws is considered. Road networks are represented as graphs composed by arcs that meet at some junctions. The crucial point is given by junctions, where interactions occur and the problem is underdetermined. The approximation of scalar conservation laws along arcs is carried out by using conservative methods, such as the classical Godunov scheme and more recent kinetic schemes with the use of suitable boundary conditions at junctions. Riemann problems are solved by means of a simulation algorithm tool. We present the algorithm and its application to some test cases and to some areas of urban network. Shock reflection for a Chaplygin gas Denis Serre UMR CNRS 5669, E.N.S. Lyon, France Denis.SERRE@umpa.ens-lyon.fr Chaplygin gases are those for which even the pressure waves correspond to linearly degenerate fields. Their numerous special features allow us to describe accurately multi dimensional shocks and their interaction. In particular, we prove the existence to some 2-D Riemann problems, including the reflection against a solid wedge. In the latter problem, the regime where the Regular Reflection takes place is determined explicitly. 6

7 Orientational aggregation and pattern formation due to local cell interaction Angela Stevens Max Planck Institute, Leipzig and University of Heidelberg, Germany In this talk kinetic equations and hyperbolic models for alignment and pattern formation in locally interacting cell systems and filaments are discussed. Possible mechanisms which allow for the selection of a finite number of orientations as well as an equal distribution of mass will be analyzed. Once alignment into a finite number of directions has taken place, questions of of pattern formation arise. Conditions for non-turing type instabilities and the selection of a wave number for the respective linearized models will be given. The talk summarizes mathematical results of several papers. Coauthors are: K. Kang, B. Perthame, I. Primi, and J.J.L. Velazquez. Transport equations and oscillations in hyperbolic equations and systems of two conservation laws Athanasios E. Tzavaras University of Maryland, U.S.A. tzavaras@math.umd.edu In this talk we apply techniques motivated by kinetic theory to problems of homogenization or issues of propagation of oscillations in systems of two conservation laws. For the homogenization problem in a context of linear transport (or transport diffusion) equations, we introduce a kinetic variable and map the homogenization problem to a hyperbolic limit type of problem for a kinetic equation. This provides a systematic framework for studying homogenization limits (without a priori knowledge of the effective equation) and for validating results obtained via multi scale asymptotic expansions. In the context of systems of two conservation laws, we will review how the transport (or semi transport) equations obtained via the kinetic formulation can be used to provide information on cancellations and coupling of oscillations. For the equations on one dimensional elastodynamics singular entropies and precise estimations on the Riemann function can be used to analyze compactness in the energy norm setting. 7

8 New vistas in stability of viscous shock and detonation waves Kevin Zumbrun Indiana University, U.S.A. Recent Lyapunov type theorems established by Mascia Zumbrun, Gues Metivier Williams Zumbrun, Howard Raoofi Zumbrun, Lyng Raoofi Texiier Zumbrun, Texier Zumbrun, and others have reduced the study of nonlinear stability, asymptotic behavior, and bifurcation of viscous traveling waves in gas dynamics combustion and MHD to determination of point spectra of the corresponding linearized operator about the wave: that is, the study of the associated eigenvalue ODE. We discuss the mathematical issues arising in this new landscape and propose general strategies for their analysis, by a balanced approach combining numerical computation, asymptotic ODE/singular perturbation theory, and classical energy estimates. 8

9 Invited talks Relaxation approximation of the Kerr model for the impedance initial boundary value problem Gilles Carbou University of Bordeaux 1, France We study different models for the electromagnetic waves propagation in an isotropic nonlinear material (a cristal for example). In the Kerr model, the constitutive relation liking the electric field E and the electric displacement D is given by D = (1 + E 2 )E. In the Kerr Debye model, the constituve relation is given by D = (1 + χ)e where χ satisfies the equation t χ = 1 ε ( E 2 χ). In fact the Kerr Debye model is a relaxation approximation of the Kerr model in which the relaxation coefficient ε is a finite response time of the nonlinear material. We consider the initial boundary value problem with null initial data in R + t Ω x, where Ω = R + R 2 is the domain in which the cristal is confined. The boundary condition is the impedance condition: H n + A ((E n) n) = ϕ, where n = ( 1, 0, 0) τ is the outer unit normal on Ω and where A is a positive endomorphism acting on Ω. For the regular solutions of this initial boundary value problem, we establish the convergence of the Kerr Debye model to the Kerr model when the relaxation coefficient ε tends to zero. 9

10 Regular reflection and the sonic criterion Volker Elling Brown University, U.S.A. Volker Reflection of shocks from a solid surface is a classical problem studied by Mach and von Neumann. There are two types of interaction, regular reflection and various forms of Mach reflection. We construct regular reflection as an exact solution, using self-similar compressible potential flow as model. A longstanding open problem is the transition point from regular to Mach reflection. We argue that the sonic criterion is the correct transition point and discuss how to construct solutions up to that point in some cases. Modeling, analysis and simulations of a tunnel fire model Ingenuin Gasser University of Hamburg, Germany gasser@math.uni-hamburg.de We discuss how to model fire events in long vehicular or railway tunnels. A major problem lies in the fact that one one hand the (air/smoke) flow has very low Mach number and on the other hand an incompressible descriptions is inappropriate. We derive a one dimensional dynamic fluid dynamic model. We study well posedness of the model and discuss the long time dynamics. Finally, numerical simulations are presented to underline the validity of the model. Systems of 1-D conservation laws with geometric constraints Helge Kristian Jenssen Penn State University, U.S.A. jenssen@math.psu.edu We consider the problem of constructing systems of one-dimensional conservation laws for which part of their geometric structure is prescribed. E.g., we might be given n linearly independent vector fields on R n, and then ask for the possible maps f such that the system u t + f(u) x = 0 has the given vector fields as its eigenvectors. This problem typically yields an overdetermined system of PDEs for the corresponding eigenvalues. Alternative problems are: to find systems with a given set of eigenvalues, or, much harder, to find systems with prescribed Hugoniot loci. We will consider examples and investigate the roles of overdeterminacy, strict hyperbolicity, existence of coordinate systems of Riemann invariants etc. The work is joint with Irina Kogan (North Carolina State University). 10

11 Shocks in thin films Andreas Münch Humboldt University, Berlin, Germany In this talk, I will discuss several examples from my work on thin film problems relating to shock profiles. This will include undercompressive waves that appear in Marangonigravity driven films and their interaction with the two types of meniscus solutions that connect the thin film with the reservoir from which it rises. Secondly, in a related problem I will discuss some ongoing work on further shock waves that appear in the Landau-Levich dragout problem when the substrate is partially wetting. Finite energy solutions to the isentropic Euler equations with geometric effects Michael Westdickenberg Georgia Institute of Technology, U.S.A. mwest@math.gatech.edu We prove the existence of spherically symmetric entropy solutions to the isentropic Euler equations, with the singularity at the origin included. We develop a compactness framework for sequences of solutions that does not require uniform boundedness. Instead we exclusively rely on the natural bounds of finite mass and total energy. Joint work with Philippe G. LeFloch. 11