Showing papers on "Hyperbolic partial differential equation published in 2000"

Control Theory for Partial Differential Equations: Continuous and Approximation Theories

Abstract: Originally published in 2000, this is the second volume of a comprehensive two-volume treatment of quadratic optimal control theory for partial differential equations over a finite or infinite time horizon, and related differential (integral) and algebraic Riccati equations. Both continuous theory and numerical approximation theory are included. The authors use an abstract space, operator theoretic approach, which is based on semigroups methods, and which unifies across a few basic classes of evolution. The various abstract frameworks are motivated by, and ultimately directed to, partial differential equations with boundary/point control. Volume 2 is focused on the optimal control problem over a finite time interval for hyperbolic dynamical systems. A few abstract models are considered, each motivated by a particular canonical hyperbolic dynamics. It presents numerous fascinating results. These volumes will appeal to graduate students and researchers in pure and applied mathematics and theoretical engineering with an interest in optimal control problems.

Mathematical Aspects of Numerical Solution of Hyperbolic Systems

Abstract: A number of physical phenomena are described by nonlinear hyperbolic equations Presence of discontinuous solutions motivates the necessity of development of reliable numerical methods based on the fundamental mathematical properties of hyperbolic systems Construction of such methods for systems more complicated than the Euler gas dynamic equations requires the investigation of existence and uniqueness of the self-similar solutions to be used in the development of discontinuity-capturing high-resolution numerical methods This frequently necessitates the study of the behavior of discontinuities under vanishing viscosity and dispersion We discuss these problems in the application to the magnetohydrodynamic equations, nonlinear waves in elastic media, and electromagnetic wave propagation in magnetics

Relationship between Symmetries andConservation Laws

Abstract: The fundamental relation between Lie-Backlund symmetry generators andconservation laws of an arbitrary differential equation is derived without regardto a Lagrangian formulation of the differential equation. This relation is used inthe construction of conservation laws for partial differential equations irrespectiveof the knowledge or existence of a Lagrangian. The relation enables one toassociate symmetries to a given conservation law of a differential equation.Applications of these results are illustrated for a range of examples.

Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation

Abstract: The aim of this article is twofold. First we consider the wave equation in the hyperbolic space HI and obtain the counterparts of the Strichartz type estimates in this context. Next we examine the relationship between semilinear hyperbolic equations in the Minkowski space and in the hyperbolic space. This leads to a simple proof of the recent result of Georgiev, Lindblad and Sogge on global existence for solutions to semilinear hyperbolic problems with small data. Shifting the space-time Strichartz estimates from the hyperbolic space to the Minkowski space yields weighted Strichartz estimates in Rn x 1R which extend the ones of Georgiev, Lindblad, and Sogge.

A Third-Order Semidiscrete Central Scheme for Conservation Laws and Convection-Diffusion Equations

Abstract: We present a new third-order, semidiscrete, central method for approximating solutions to multidimensional systems of hyperbolic conservation laws, convection-diffusion equations, and related problems. Our method is a high-order extension of the recently proposed second-order, semidiscrete method in [A. Kurgonov and E. Tadmor, J. Comput Phys., 160 (2000) pp. 241--282]. The method is derived independently of the specific piecewise polynomial reconstruction which is based on the previously computed cell-averages. We demonstrate our results by focusing on the new third-order central weighted essentially nonoscillatory (CWENO) reconstruction presented in [D. Levy, G. Puppo, and G. Russo, SIAM J. Sci. Comput., 21 (1999), pp. 294--322]. The numerical results we present show the desired accuracy, high resolution, and robustness of our method.

A Particle-Partition of Unity Method for the Solution of Elliptic, Parabolic, and Hyperbolic PDEs

Abstract: In this paper, we present a meshless discretization technique for instationary convection-diffusion problems. It is based on operator splitting, the method of characteristics, and a generalized partition of unity method. We focus on the discretization process and its quality. The method may be used as an h-version or a p-version. Even for general particle distributions, the convergence behavior of the different versions corresponds to that of the respective version of the finite element method on a uniform grid. We discuss the implementational aspects of the proposed method. Furthermore, we present the results of numerical examples, where we considered instationary convection-diffusion, instationary diffusion, linear advection, and elliptic problems.

Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation

Abstract: The aim of this article is threefold. First, we use the FBI transform to set up a calculus for partial differential operators with nonsmooth coefficients. Next, this calculus allows us to prove Strichartz type estimates for the wave equation with nonsmooth coefficients. Finally, we use these Strichartz estimates to improve the local theory for second order nonlinear hyperbolic equations. 1. Introduction. The FBI transform is, in a way, similar to the complex Fourier transform, in that for each function in Rn it provides a representation as a holomorphic function in R2n. However, in the case of the FBI transform we can identify naturally R2n with the phase space T*Rn. For a pseudodifferential operator with smooth symbol acting on functions in Rn one can produce by conjugation a corresponding formal series acting on func tions in R2n9 for which the first term is exactly the multiplication by the symbol. This series converges and has a nice representation in the Weyl calculus provided that the symbol of the operator is analytic. This is how the FBI transform has been used in the study of partial differential operators with analytic coefficients; see (12), (13), where this machinery is developed. Here, in a way, we do the opposite: we look at operators with nonsmooth coefficients, approximate the conjugated operator by a partial sum of the formal series, and prove error estimates. In the simplest case the approximate conjugate operator is exactly the multiplication by the symbol. This is also related to the Cordoba-Fefferman wave-packet transform in (3). In the third section we use the error estimates to reduce the Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients to weighted LP(Lq) ?> L2 estimates for the FBI transform. These, in turn, can be proved in the usual fashion, using appropriate oscillatory integral esti mates. In the last part of the article we explain how one can use the new Strichartz type estimates to improve the local theory for nonlinear hyperbolic equations beyond the classical setup. These results are not sharp and will be improved in subsequent articles.

Shadowing in Dynamical Systems: Theory and Applications

Abstract: Preface. 1. Hyperbolic Fixed Points of Diffeomorphisms and Their Stable and Unstable Manifolds. 2. Hyperbolic Sets of Diffeomorphisms. 3. Transversal Homoclinic Points of Diffeomorphisms and Hyperbolic Sets. 4. The Shadowing Theorem for Hyperbolic Sets of Diffeomorphisms. 5. Symbolic Dynamics Near a Transversal Homoclinic Point of a Diffeomorphism. 6. Hyperbolic Periodic Orbits of Ordinary Differential Equations, Stable and Unstable Manifolds and asymptotic Phase. 7. Hyperbolic Sets of Ordinary Differential Equations. 8. Transversal Homoclinic Points and Hyperbolic Sets in Differential Equations. 9. Shadowing Theorems for Hyperbolic Sets of Differential Equations. 10. Symbolic Dynamics Near a Transversal Homoclinic Orbit of a System of Ordinary Differential Equations. 11. Numerical Shadowing. References.

Stabilized hp -Finite Element Methods for First-Order Hyperbolic Problems

Abstract: We analyze the hp-version of the streamline-diffusion finite element method (SDFEM) and of the discontinuous Galerkin finite element method (DGFEM) for first-order linear hyperbolic problems. For both methods, we derive new error estimates on general finite element meshes which are sharp in the mesh-width h and in the spectral order p of the method, assuming that the stabilization parameter is O(h/p). For piecewise analytic solutions, exponential convergence is established on quadrilateral meshes. For the DGFEM we admit very general irregular meshes and for the SDFEM we allow meshes which contain hanging nodes. Numerical experiments confirm the theoretical results.

Nonlinear stochastic wave and heat equations

Abstract: We study nonlinear wave and heat equations on ℝ d driven by a spatially homogeneous Wiener process. For the wave equation we consider the cases of d = 1, 2, 3. The heat equation is considered on an arbitrary ℝ d -space. We give necessary and sufficient conditions for the existence of a function-valued solution in terms of the covariance kernel of the noise.

Centred TVD schemes for hyperbolic conservation laws

Abstract: New first- and high-order centred methods for conservation laws are presented. Convenient TVD conditions for constructing centred TVD schemes are then formulated and some useful results are proved. Two families of centred TVD schemes are constructed and extended to nonlinear systems. Some numerical results are also presented.

Inertial waves in a rotating spherical shell: attractors and asymptotic spectrum

Abstract: We investigate the asymptotic properties of inertial modes confined in a spherical shell when viscosity tends to zero. We first consider the mapping made by the characteristics of the hyperbolic equation (Poincare's equation) satisfied by inviscid solutions. Characteristics are straight lines in a meridional section of the shell, and the mapping shows that, generically, these lines converge towards a periodic orbit which acts like an attractor. We then examine the relation between this characteristic path and eigensolutions of the inviscid problem and show that in a purely two-dimensional problem, convergence towards an attractor means that the associated velocity field is not square-integrable. We give arguments which generalize this result to three dimensions. We then consider the viscous problem and show how viscosity transforms singularities into internal shear layers which in general betray an attractor expected at the eigenfrequency of the mode. We find that there are nested layers, the thinnest and most internal layer scaling with $E^{1/3}$-scale, $E$ being the Ekman number. Using an inertial wave packet traveling around an attractor, we give a lower bound on the thickness of shear layers and show how eigenfrequencies can be computed in principle. Finally, we show that as viscosity decreases, eigenfrequencies tend towards a set of values which is not dense in $[0,2\Omega]$, contrary to the case of the full sphere ($\Omega$ is the angular velocity of the system). Hence, our geometrical approach opens the possibility of describing the eigenmodes and eigenvalues for astrophysical/geophysical Ekman numbers ($10^{-10}-10^{-20}$), which are out of reach numerically, and this for a wide class of containers.

Numerical Schemes for Hyperbolic Systems of Conservation Laws with Stiff Diffusive Relaxation

Abstract: Hyperbolic systems of conservation laws often have diffusive relaxation terms that lead to a small relaxation limit governed by reduced systems of a parabolic or hyperbolic type. In such systems the understanding of basic wave pattern is difficult to achieve, and standard high resolution methods fail to describe the right asymptotic behavior unless the small relaxation rate is numerically resolved. We develop high resolution underresolved numerical schemes that possess the discrete analogue of the continuous asymptotic limit, which are thus able to approximate the equilibrium system with high order accuracy even if the limiting equations may change type.

On the diffusion phenomenonof quasilinear hyperbolic waves

Abstract: We consider the asymptotic behavior of solutions of the quasilinear hyperbolic equation with linear damping u tt +u t − div (a(∇u)∇u)=0, and show that they tend, as t→+∞ , to those of the nonlinear parabolic equation v t − div (a(∇v)∇v)=0, in the sense that the norm ‖u(. ,t)−v(. ,t)‖ L ∞ ( R n ) of the difference u−v decays faster than that of either u or v . This provides another example of the diffusion phenomenon of nonlinear hyperbolic waves, first observed by L. Hsiao and Tai-ping Liu.

A Moving Mesh Method for One-dimensional Hyperbolic Conservation Laws

Abstract: We develop an adaptive method for solving one-dimensional systems of hyperbolic conservation laws that employs a high resolution Godunov-type scheme for the physical equations, in conjunction with a moving mesh PDE governing the motion of the spatial grid points. Many other moving mesh methods developed to solve hyperbolic problems use a fully implicit discretization for the coupled solution-mesh equations, and so suffer from a significant degree of numerical stiffness. We employ a semi-implicit approach that couples the moving mesh equation to an efficient, explicit solver for the physical PDE, with the resulting scheme behaving in practice as a two-step predictor-corrector method. In comparison with computations on a fixed, uniform mesh, our method exhibits more accurate resolution of discontinuities for a similar level of computational work.

Central Schemes for Balance Laws of Relaxation Type

Abstract: Several models in mathematical physics are described by quasi-linear hyperbolic systems with source term and in several cases the production term can become stiff. Here suitable central numerical schemes for such problems are developed and applications to the Broadwell model and extended thermodynamics are presented. The numerical methods are a generalization of the Nessyahu--Tadmor scheme to the nonhomogeneous case by including the cell averages of the production terms in the discrete balance equations. A second order scheme uniformly accurate in the relaxation parameter is derived and its properties analyzed. Numerical tests confirm the accuracy and robustness of the scheme.

Discontinuous hp-finite element methods for advection-diffusion problems

Abstract: We consider the hp-version of the discontinuous Galerkin finite element method for second-order partial differential equations with nonnegative characteristic form. This class of equations includes second--order elliptic and parabolic equations, first-order hyperbolic equations, as well as problems of mixed hyperbolic-elliptic-parabolic type. Our main concern is the error analysis of the method in the absence of streamline-diffusion stabilization. In the hyperbolic case, an hp-optimal error bound is derived. In the self-adjoint elliptic case, an error bound that is h-optimal and p-suboptimal by half a power of p is obtained. These estimates are then combined to deduce an error bound in the general case. For element-wise analytic solutions the method exhibits exponential rates of convergence under p-refinement. The theoretical results are illustrated by numerical experiments.

Compressible Euler equations with general pressure law

Abstract: We study the hyperbolic system of Euler equations for an isentropic, compressible fluid governed by a general pressure law. The existence and regularity of the entropy kernel that generates the family of weak entropies is established by solving a new Euler-Poisson-Darboux equation, which is highly singular when the density of the fluid vanishes. New properties of cancellation of singularities in combinations of the entropy kernel and the associated entropy-flux kernel are found.

Initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws

Abstract: We consider initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws through the scalar case. The entropy solutions we address assume their initial data only in the sense of weak-star in L ∞ as t→0+ and satisfy the entropy inequality in the sense of distributions for t>0. We prove that, if the flux function has weakly genuine nonlinearity, then the entropy solutions are always unique and the initial layers do not appear. We also discuss applications to the zero relaxation limit for hyperbolic systems of conservation laws with relaxation.

Global existence of classical solutions to systems of wave equations with critical nonlinearity in three space dimensions

Abstract: We discuss the global existence of small solutions to the Cauchy problem for systems of quasilinear wave equations in three space dimensions, when their nonlinear terms have quadratic nonlinearity. A global existence theorem is established on the null condition which is extended to the condition for systems of wave equations with different propagation speeds.

Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms

Abstract: Here a; b?0 and p?1, m?1. In case of IBVP, in a bounded domain ⊂Rn with Dirichlet boundary conditions, the following results are known: 1. When a=0, it is proved (see [1, 3, 8, 14, 16]) that the solution blows up in nite time for su ciently large initial data. 2. When b=0; Haraux and Zuazua [5] and Kopackova [7] prove the global existence result for large initial data. The behavior of the solution of Eq. (1.1) with nonlinear source and linear damping (case m=1) in an abstract setting was considered by Levine in [9]. More precisely, he showed that the solutions with negative initial energy are nonglobal.

Viability property for a backward stochastic differential equation and applications to partial differential equations

Abstract: In the present paper, we study conditions under which the solutions of a backward stochastic differential equation remains in a given set of constraints. This property is the so-called “viability property”. In a separate section, this condition is translated to a class of partial differential equations.

Evolution Galerkin methods for hyperbolic systems in two space dimensions

Abstract: , The subject of the paper is the analysis of three new evolution Galerkin schemes for a system of hyperbolic equations, and particularly for the wave equation system. The aim is to construct methods which take into account all of the infinitely many directions of propagation of bicharacteristics. The main idea of the evolution Galerkin methods is the following: the initial function is evolved using the characteristic cone and then projected onto a finite element space. A numerical comparison is given of the new methods with already existing methods, both those based on the use of bicharacteristics as well as commonly used finite difference and finite volume methods. We discuss the stability properties of the schemes and derive error estimates.