Sup-norm stability for Glimm's scheme

We consider the Cauchy problem for a general N × N system of conservation laws. Existence of solutions was proved by Glimm using his celebrated random choice scheme. In this paper, we obtain a third-order interaction estimate analagous to that obtained by Glimm for 2×2 systems. By using this estimate, and identifying a global cancellation effect, we obtain L^∞-stability for solutions generated by Glimm's scheme. As an immediate consequence we have L¹-stability and L^∞-decay, obtained by Temple for 2×2 systems. © 1993 John Wiley & Sons, Inc.     

Shock Layers Interactions for a Relaxation Approximation to Conservation Laws

We consider the interaction between shock waves in a semilinear relaxation approximation to N×N systems of conservation laws. This interaction is described by proving the existence of a particular solution to the relaxing system, with a certain asymptotic as Received July 1998     

Entropy flux-splittings for hyperbolic conservation laws part I: General framework

A general framework is proposed for the derivation and analysis of flux-splittings and the corresponding flux-splitting schemes for systems of conservation laws endowed with a strictly convex entropy. The approach leads to several new properties of the existing flux-splittings and to a method for the construction of entropy flux-splittings for general situations. A large family of genuine entropy flux-splittings is derived for several significant examples: the scalar conservation laws, the p-system, and the Euler system of isentropic gas dynamics. In particular, for the isentropic Euler system, we obtain a family of splittings that satisfy the entropy inequality associated with the mechanical energy. For this system, it is proved that there exists a unique genuine entropy flux-splitting that satisfies all of the entropy inequalities, which is also the unique diagonalizable splitting. This splitting can be also derived by the so-called kinetic formulation. Simple and useful difference schemes are derived from the flux-splittings for hyperbolic systems. Such entropy flux-splitting schemes are shown to satisfy a discrete cell entropy inequality. For the diagonalizable splitting schemes, an a priori L^∞ estimate is provided by applying the principle of bounded invariant regions. The convergence of entropy flux-splitting schemes is proved for the 2 × 2 systems of conservation laws and the isentropic Euler system. ©1995 John Wiley & Sons, Inc.     

An instability of the Godunov scheme

We construct a solution to a 2 × 2 strictly hyperbolic system of conservation laws, showing that the Godunov scheme [13] can produce an arbitrarily large amount of oscillations. This happens when the speed of a shock is close to rational, inducing a resonance with the grid. Differently from the Glimm scheme or the vanishing‐viscosity method, for systems of conservation laws our counterexample indicates that no a priori BV bounds or L¹‐stability estimates can in general be valid for finite difference schemes. © 2006 Wiley Periodicals, Inc.     

Analysis of a circulant based preconditioner for a class of lower rank extracted systems

This paper proposes and studies the performance of a preconditioner suitable for solving a class of symmetric positive definite systems, A_px=b, which we call lower rank extracted systems (LRES), by the preconditioned conjugate gradient method. These systems correspond to integral equations with convolution kernels defined on a union of many line segments in contrast to only one line segment in the case of Toeplitz systems. The p × p matrix, A_p, is shown to be a principal submatrix of a larger N × N Toeplitz matrix, A_N. The preconditioner is provided in terms of the inverse of a 2N × 2N circulant matrix constructed from the elements of A_N. The preconditioner is shown to yield clustering in the spectrum of the preconditioned matrix similar to the clustering results for iterative algorithms used to solve Toeplitz systems. The analysis also demonstrates that the computational expense to solve LRE systems is reduced to O(N log N). Copyright © 2004 John Wiley & Sons, Ltd.     

Multidimensional viscous shocks II: The small viscosity limit

In this paper we prove the existence of curved, multidimensional viscous shocks and also justify the small‐viscosity limit. Starting with a curved, multidimensional (inviscid) shock solution to a system of hyperbolic conservation laws, we show that the shock can be obtained as a small‐viscosity limit of solutions to an associated parabolic problem (viscous shocks). The two main hypotheses are a natural Evans function assumption on the viscous profile, together with a restriction on how much the shock can deviate from flatness. The main tools are a conjugation lemma that removes x_N/? dependence from the linearization of the parabolic problem about the viscous profile, new degenerate Kreiss‐type symmetrizers used to prove an L² estimate for the linearized problem, and a finite‐regularity calculus of semiclassical and mixed type (classical‐semiclassical) pseudodifferential operators. © 2003 Wiley Periodicals, Inc.     

The Geometry of 2 × 2 Systems of Conservation Laws

We consider one typical two-parameter family of quadratic systems of 2 × 2 conservation laws, and study the geometry of the behaviour of the possible solutions of the Riemann problem near an umbilic point, following the geometric approach presented by Isaacson, Marchesin, Palmeira, Plohr, in A global formalism for nonlinear waves in conservation laws, Commun. Math. Phys. (1992). The corresponding phase portraits for the rarefaction curves, shock curves and composite curves are discussed. Financial support from FCT and Calouste Gulbenkian Foundation.     

Two-Wave Interactions for Weakly Nonlinear Hyperbolic Waves

This paper is devoted to studying the weakly nonlinear interaction of two waves whose propagation is governed by n × n hyperbolic systems of conservation laws. Our method of approach involves introducing two nonlinear phase variables and carrying out a perturbation analysis. This extended version of our previous single-wave-mode theory [5] is then applied to the equations of gas dynamics to study interacting sound waves. Numerical results for the wave-wave interaction are presented graphically in a set of figures.     

Blow up for a class of quasilinear wave equations in one space dimension

For suitable σ and F, we prove that all classical solutions of the quasilinear wave equation , with initial data of compact support, develop singularities in finite time. The assumptions on σ and F include in particular the model case , for q ⩾ 2,and ϵ = ±1. The starting point of the proof is to write the equation under the form of a first order system of two equations, in which F(ϕ) appears as a nonlocal term. Then, we present a new idea to control the effect of this perturbation term, and we conclude the proof by using well‐known methods developed for 2 × 2 systems of conservation laws. Copyright © 2000 John Wiley & Sons, Ltd.     

Wiener-hopf equations for waves scattered by a system of parallel sommerfeld half-planes

A scalar time-harmonic wave (governed by Helmholtz's equation) impinges on N semi-infinite half-planes. The scattered field is sought when first, second, and third-kind boundary conditions or even general linear transmission conditions on the plates ∑_m and their complementary parts ∑ are prescribed. Making use of the Fourier transform a representation formula for H¹ (Ω) solutions is presented. The boundary/transmission problem is shown to be equivalent to a (2N × 2N)-Wiener–Hopf (WH) system for jumps of the Dirichlet–and Neumann–Cauchy data across the semi-infinite screens ∑_m. The (2N × 2N)-Fourier symbol matrix ??_?? contains N block matrices on the diagonal corresponding to Sommerfeld boundary/transmission problems for a single plate. These (2 × 2)-symbol matrices are factorizable and thus the full WH system is invertible by a perturbation argument for not too small spacings of neighbouring screens ∑_m.     

Boundary value problem for the N‐dimensional time periodic Vlasov–Poisson system

In this work, we study the existence of time periodic weak solution for the N‐dimensional Vlasov–Poisson system with boundary conditions. We start by constructing time periodic solutions with compact support in momentum and bounded electric field for a regularized system. Then, the a priori estimates follow by computations involving the conservation laws of mass, momentum and energy. One of the key point is to impose a geometric hypothesis on the domain: we suppose that its boundary is strictly star‐shaped with respect to some point of the domain. These results apply for both classical or relativistic case and for systems with several species of particles. Copyright © 2006 John Wiley & Sons, Ltd.     

Antidiffusive Lagrangian‐remap schemes for models of polydisperse sedimentation

One‐dimensional models of gravity‐driven sedimentation of polydisperse suspensions with particles that belong to N size classes give rise to systems of N strongly coupled, nonlinear first‐order conservation laws for the local solids volume fractions. As the eigenvalues and eigenvectors of the flux Jacobian have no closed algebraic form, characteristic‐wise numerical schemes for these models become involved. Alternative simple schemes for this model directly utilize the velocity functions and are based on splitting the system of conservation laws into two different first‐order quasi‐linear systems, which are solved successively for each time iteration, namely, the Lagrangian and remap steps (so‐called Lagrangian‐remap [LR] schemes). This approach was advanced in (Bürger, Chalons, and Villada, SIAM J Sci Comput 35 (2013), B1341–B1368) for a multiclass Lighthill–Whitham‐Richards traffic model with nonnegative velocities. By incorporating recent antidiffusive techniques for transport equations a new version of these Lagrangian‐antidiffusive remap (L‐AR) schemes for the polydisperse sedimentation model is constructed. These L‐AR schemes are supported by a partial analysis for N = 1. They are total variation diminishing under a suitable CFL condition and therefore converge to a weak solution. Numerical examples illustrate that these schemes, including a more accurate version based on MUSCL extrapolation, are competitive in accuracy and efficiency with several existing schemes. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1109–1136, 2016     

A least-squares finite element method based on the Helmholtz decomposition for hyperbolic balance laws

In this paper, a least-squares finite element method for scalar nonlinear hyperbolic balance laws is proposed and studied. The approach is based on a formulation that utilizes an appropriate Helmholtz decomposition of the flux vector and is related to the standard notion of a weak solution. This relationship, together with a corresponding connection to negative-norm least-squares, is described in detail. As a consequence, an important numerical conservation theorem is obtained, similar to the famous Lax–Wendroff theorem. The numerical conservation properties of the method in this paper do not fall precisely in the framework introduced by Lax and Wendroff, but they are similar in spirit as they guarantee that when L² convergence holds, the resulting approximations approach a weak solution to the hyperbolic problem. The least-squares functional is continuous and coercive in an H⁻¹-type norm, but not L²-coercive. Nevertheless, the L² convergence properties of the method are discussed. Convergence can be obtained either by an explicit regularization of the functional, that provides control of the L² norm, or by properly choosing the finite element spaces, providing implicit control of the L² norm. Numerical results for the inviscid Burgers equation with discontinuous source terms are shown, demonstrating the L² convergence of the obtained approximations to the physically admissible solution. The numerical method utilizes a least-squares functional, minimized on finite element spaces, and a Gauss–Newton technique with nested iteration. We believe that the linear systems encountered with this formulation are amenable to multigrid techniques and combining the method with adaptive mesh refinement would make this approach an efficient tool for solving balance laws (this is the focus of a future study).     

THE NONLINEAR STABILITY OF TRAVELLING WAVE SOLUTIONS FOR A REACTING FLOW MODEL WITH SOURCE TERM

1IntroductionConsidertilefollowingsystemtlwiwasproposedin[91todescribereactinggasinwhichthereealsttwo"lodes.Where,pcistiledensityofthemajormodeandpsisOftheminormode,f s=1.uisthevelocity,andp=PC'(r as)ispressurewhichcanbederivedbyAvogadro'sLaw.Here,cisthesoundspeedofthemajormode.TheParameterPprovidessometenuouslinkwithrealphysics.Sisthesollrcetermwhere,risareactiontime,rE(p)alldSE(P)areequilibriumdistributions.Thereaderisrefere(lto[9]formorephysicsandnumericalbackgroulld.TheLagrangianform…     

Generalized factorization for N×N Daniele–Khrapkov matrix functions

A generalization to N×N of the 2×2 Daniele–Khrapkov class of matrix‐valued functions is proposed. This class retains some of the features of the 2×2 Daniele–Khrapkov class, in particular, the presence of certain square‐root functions in its definition. Functions of this class appear in the study of finite‐dimensional integrable systems. The paper concentrates on giving the main properties of the class, using them to outline a method for the study of the Wiener–Hopf factorization of the symbols of this class. This is done through examples that are completely worked out. One of these examples corresponds to a particular case of the motion of a symmetric rigid body with a fixed point (Lagrange top). Copyright © 2001 John Wiley & Sons, Ltd.     

A two-dimensional hyperbolic system of nonlinear conservation laws with functional solutions

A two-dimensional hyperbolic system of nonlinear conservation laws is considered for any piecewise constant initial data having two discontinuity rays with the origin as vertex. One kind of new waves, which is labeled the Dirac-contact wave, appears in the solution. The entropy conditions for the Dirac-contact waves are given. The solutions on the Dirac-contact waves can be viewed as the bounded linear functionals onC ₀ ^∞ (R ² ×R ₊). Supported by CNSF and a grant from Academia Sinica Author’s current address: CMAP, Ecole Polytechnique, 91128 Palaiseau Cedex, France     

Symmetry analysis of the charged squashed Kaluza–Klein black hole metric

In this research article, a complete analysis of symmetries and conservation laws for the charged squashed Kaluza–Klein black hole space‐time in a Riemannian space is discussed. First, a comprehensive group analysis of the underlying space‐time metric using Lie point symmetries is presented, and then the n‐dimensional optimal system of this space‐time metric, for n = 1,…,4, are computed. It is shown that there is no any n‐dimensional optimal system of Lie symmetry subalgebra associated to the system of geodesic for n≥5. Then the point symmetries of the one‐parameter Lie groups of transformations that leave invariant the action integral corresponding to the Lagrangian that means Noether symmetries are found, and then the conservation laws associated to the system of geodesic equations are calculated via Noether's theorem. Copyright © 2015 John Wiley & Sons, Ltd.     

A Synthetical Two‐Component Model with Peakon Solutions

A generalized two‐component model with peakon solutions is proposed in this paper. It allows an arbitrary function to be involved in as well as including some existing integrable peakon equations as special reductions. The generalized two‐component system is shown to possess Lax pair and infinitely many conservation laws. Bi‐Hamiltonian structures and peakon interactions are discussed in detail for typical representative equations of the generalized system. In particular, a new type of N‐peakon solution, which is not in the traveling wave type, is obtained from the generalized system.     

Geometry of conservation laws for a class of parabolic partial differential equations

I consider the problem of computing the space of conservation laws for a second-order parabolic partial differential equation for one function of three independent variables. The PDE is formulated as an exterior differential system on a 12-manifold M, and its conservation laws are identified with the vector space of closed 3-forms in the infinite prolongation of modulo the so-called "trivial" conservation laws. I use the tools of exterior differential systems and Cartan's method of equivalence to study the structure of the space of conservation laws. My main result is: Theorem. Any conservation law for a second-order parabolic PDE for one function of three independent variables can be represented by a closed 3-form in the differential ideal ${\cal I}$ on the original 12-manifold M. I show that if a nontrivial conservation law exists, then has a deprolongation to an equivalent system on a 7-manifold N, and any conservation law for can be expressed as a closed 3-form on N that lies in . Furthermore, any such system in the real analytic category is locally equivalent to a system generated by a (parabolic) equation of the formA (u _xx u _yy -u ² _xy )+Bu _xx +2Cu _xy +Du _yy +E = 0 where A, B, C, D, E are functions of x, y, t, u, u _x , u _y , u _t . I compute the space of conservation laws for several examples, and I begin the process of analyzing the general case using Cartan's method of equivalence. I show that the non-linearizable equation has an infinite-dimensional space of conservation laws. This stands in contrast to the two-variable case, for which Bryant and Griffiths showed that any equation whose space of conservation laws has dimension 4 or more is locally equivalent to a linear equation, i.e., is linearizable.