Topology of elementary waves for mixed-type systems of conservation laws (original) (raw)
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Shock Wave Admissibility for Quadratic Conservation Laws
Journal of Differential Equations, 1995
In this work we present a new approach to the study of the stability of admissible shock wave solutions for systems of conservation laws that change type. The systems we treat have quadratic ux functions. We employ the fundamental wave manifold W as a global framework to characterize shock waves that comply with the viscosity admissibility criterion. Points of W parametrize dynamical systems associated with shock wave solutions. The region of W comprising admissible shock waves is bounded by the loci of structurally unstable dynamical systems. Explicit formulae are presented for the loci associated with saddle-node, Hopf, and Bogdanov-Takens bifurcation, and with straight-line heteroclinic connections. Using Melnikov's integral analysis, we calculate the tangent to the homoclinic part of the admissibility boundary at Bogdanov-Takens points of W. Furthermore, using numerical methods, we explore the heteroclinic loci corresponding to curved connecting orbits and the complete homoclinic locus. We nd the region of admissible waves for a generic, two-dimensional slice of the fundamental wave manifold, and compare it with the set of shock points that comply with the Lax admissibility criterion, thereby elucidating how this criterion di ers from viscous pro le admissibility.
Transitional Waves for Conservation Laws
SIAM Journal on Mathematical Analysis, 1990
A new class of fundamental -waves arises in conservation laws that are not strictly hyperbolic. These waves serve as transitions between wave groups associated with particular characteristic families. Transitional shock waves are discontinuous solutions that possess viscous profiles but do not conform to the Lax characteristic criterion; they are sensitive to the precise form of the physical viscosity. Transitional rarefaction waves are rarefaction fans across which the characteristic family changes from faster to slower. /' . rl this paper we4den4i <an extensive family of transitional shock waves for conservation laws with quadratic fluxes and arbitrary viscosity matrices; this family comprises all transitional shock waves for a certain class of such quadratic models. We also establish, for general systems of two conservation laws, the generic nature of rarefaction curves near an elliptic region, thereby identifying transitional rarefaction waves. The use of transitional waves in solving Riemann problems is illustrated in an example where the characteristic and viscous profile admissibility criteria yield distinct solutions. (k __._ AMS (MOS) Subject Classification: 34D30, 35L65, 35L67, 35L80, 58F09
Topological resolution of Riemann problems for pairs of conservation laws
Quarterly of Applied Mathematics, 2010
The structure of Riemann solutions for certain systems of conservation laws can be so complicated that the classical constructions are unable to establish global existence and stability. For systems of two conservation laws, classically the local solution is found by intersecting two wave curves specified by the Riemann data. The intersection point 2000 Mathematics Subject Classification. 35L65.
Degenerate systems of conservation laws
Contemporary Mathematics, 1987
We describe;-systems of conservation laws with the property that the shock and rarefaction curves coincide. 4W give ,new examples of such systems. These systems isolate and separate many nonlinear aspects of shock waves. The author believes that analytical techniques which handle problems of uniqueness, continuous dependence or convergence of finite difference schemes in these systems, would isolate components in a corresponding analysis required for general systems of conservation laws. .
Lax Shocks in Mixed-Type Systems of Conservation Laws
Journal of Hyperbolic Differential Equations, 2008
Small amplitude shocks involving a state with complex characteristic speeds arise in mixed-type systems of two or more conservation laws. We study such shocks in detail in the generic case, when they appear near the codimension-1 elliptic boundary. Then we classify all exceptional codimension-2 states on smooth parts of the elliptic boundary. Asymptotic formulae describing shock curves near regular and exceptional states are derived. The type of singularity at the exceptional point depends on the second and third derivatives of the flux function. The main application is understanding the structure of small amplitude Riemann solutions where one of the initial states lies in the elliptic region.
Nonclassical Shock Waves of Conservation Laws: Flux Function Having Two Inflection Points
2006
We consider the Riemann problem for non-genuinely nonlinear conservation laws where the flux function admits two inflection points. This is a simplification of van der Waals fluid pressure, which can be seen as a function of the specific volume for a specific entropy at which the system lacks the non-genuine nonlinearity. Corresponding to each inflection point, A nonclassical Riemann solver
Classification of homogeneous quadratic conservation laws with viscous terms
Computational & Applied Mathematics, 2007
In this paper, we study systems of two conservation laws with homogeneous quadratic flux functions. We use the viscous profile criterion for shock admissibility. This criterion leads to the occurrence of non-classical transitional shock waves, which are sensitively dependent on the form of the viscosity matrix. The goal of this paper is to lay a foundation for investigating how the structure of solutions of the Riemann problem is affected by the choice of viscosity matrix. Working in the framework of the fundamental wave manifold, we derive a necessary and sufficient condition on the model parameters for the presence of transitional shock waves. Using this condition, we are able to identify the regions in the wave manifold that correspond to transitional shock waves. Also, we determine the boundaries in the space of model parameters that separate models with differing numbers of transitional regions.
Transonic Shock Formation in a Rarefaction Riemann Problem for the 2D Compressible Euler Equations
SIAM Journal on Applied Mathematics, 2008
It is perhaps surprising for a shock wave to exist in the solution of a rarefaction Riemann problem for the compressible Euler equations in two space dimensions. We present numerical evidence and generalized characteristic analysis to establish the existence of a shock wave in such a 2D Riemann problem, defined by the interaction of four rarefaction waves. We consider both the customary configuration of waves at the right angle and also an oblique configuration for the rarefaction waves. Two distinct mechanisms for the formation of a shock wave are discovered as the angle between the waves is varied. between waves may be considered and special solutions (stationary wave interactions) in general will occur at angles other than 90 o , see . From the point of view of defining a Riemann solution for a finite difference mesh, we might consider a variety of meshes with different angles between the cell edges. In accordance with both points of view, we consider the oblique four-wave Riemann problem. We perform refined numerical experiments, using the FronTier code developed at the AMS department, SUNY Stony Brook and obtain resolved numerical solutions. This code uses a five point vectorized split MUSCL scheme [4] as a shock capturing algorithm. It is second order accurate for smooth solutions and first order accurate near shock waves. We solved the full compressible Euler equations in the original x, y, t coordinates, not in self-similar coordinates, so the numerics are actually very well documented in previous literature [4] and . Our main result is the existence of shock wave, established numerically by several different criteria, for a 2D Riemann problem with four rarefaction waves in both the 90 degrees case and the oblique case. The possibility of shock formation indicates the deep sophistication of this seemingly easy problem. We formulate plausible structures for the solution via the method of generalized characteristic analysis (i. e., the analysis of characteristics, shocks, and sonic curves or the law of causality). In Section 2, we formulate the problem under study and discuss an algorithm for the construction of characteristics in the numerical solutions. The existence of shock waves is established by multiple criteria used in our numerical studies to indicate the presence of shock waves. Specifically, we consider 1. plots of density and pressure on a curve through the shocks; 2. nontangential termination of characteristics at the shock front; 3. convergence of characteristics of the same family at the local shock front; 4 pattern recognition software for automated shock wave detection. 5. stability of above criteria under mesh refinement.
Systems of conservation laws with invariant submanifolds
Transactions of the American Mathematical Society, 1983
Systems of conservation laws with coinciding shock and rarefaction curves arise in the study of oil reservoir simulation, multicomponent chromatography, as well as in the study of nonlinear motion in elastic strings. Here we characterize this phenomenon by deriving necessary and sufficient conditions on the geometry of a wave curve in order that the shock wave curve coincide with its associated rarefaction wave curve for a system of conservation laws. This coincidence is the one dimensional case of a submanifold of the state variables being invariant for the system of equations, and the necessary and sufficient conditions are derived for invariant submanifolds of arbitrary dimension. In the case of 2 × 2 2 \times 2 systems we derive explicit formulas for the class of flux functions that give rise to the coupled nonlinear conservation laws for which the shock and rarefaction wave curves coincide.
SIAM Journal on Mathematical Analysis, 2009
The evolution of discontinuity and formation of triple-shock pattern in solutions to a two-dimensional hyperbolic system of conservation laws are studied. When the initial discontinuity is a convex curve, it is discovered that the structure of the global solution changes dramatically around a critical time: After the critical time, a triple-shock pattern forms, while, before the critical time, only two shocks are developed. The envelope surface of intersections and the evolution of discontinuity are analyzed by developing new ideas and approaches. The global structure of the entropy solution is presented.