Showing papers on "Conservation law published in 2002"

Finite Volume Methods for Hyperbolic Problems

Abstract: Preface 1. Introduction 2. Conservation laws and differential equations 3. Characteristics and Riemann problems for linear hyperbolic equations 4. Finite-volume methods 5. Introduction to the CLAWPACK software 6. High resolution methods 7. Boundary conditions and ghost cells 8. Convergence, accuracy, and stability 9. Variable-coefficient linear equations 10. Other approaches to high resolution 11. Nonlinear scalar conservation laws 12. Finite-volume methods for nonlinear scalar conservation laws 13. Nonlinear systems of conservation laws 14. Gas dynamics and the Euler equations 15. Finite-volume methods for nonlinear systems 16. Some nonclassical hyperbolic problems 17. Source terms and balance laws 18. Multidimensional hyperbolic problems 19. Multidimensional numerical methods 20. Multidimensional scalar equations 21. Multidimensional systems 22. Elastic waves 23. Finite-volume methods on quadrilateral grids Bibliography Index.

Fractional Schrödinger equation.

Abstract: Some properties of the fractional Schrodinger equation are studied. We prove the Hermiticity of the fractional Hamilton operator and establish the parity conservation law for fractional quantum mechanics. As physical applications of the fractional Schrodinger equation we find the energy spectra of a hydrogenlike atom (fractional "Bohr atom") and of a fractional oscillator in the semiclassical approximation. An equation for the fractional probability current density is developed and discussed. We also discuss the relationships between the fractional and standard Schrodinger equations.

Front Tracking for Hyperbolic Conservation Laws

Abstract: In this edition we have added the following new material: In Chapt. 1 we have added a section on linear equations, which allows us to present some of the material in the book in the simpler linear setting. In Chapt. 2 we have made some changes in the presentation of Kružkov’s fundamental doubling of variables method. In Chapt. 3 on finite difference methods the focus has been changed to finite volume methods. A section on higher-order schemes has been added. The section on measure-valued solutions has been rewritten. The main existence theorem in Chapt. 4, Theorem 4.3, now resembles the one-dimensional case. The presentation of the solution of the Riemann problem for systems in Chapt. 5 has been supplemented by the complete solution of the Riemann problem for the 3 3 Euler equations of gas dynamics. The solution of the Cauchy problem for systems in Chapt. 6 has been rewritten and simplified. We have added a new chapter, Chapt. 8, on one-dimensional scalar conservation laws where the flux function depends explicitly on space in a discontinuous manner

Direct construction method for conservation laws of partial differential equations. Part I: Examples of conservation law classifications

Abstract: An effective algorithmic method is presented for finding the local conservation laws for partial differential equations with any number of independent and dependent variables. The method does not require the use or existence of a variational principle and reduces the calculation of conservation laws to solving a system of linear determining equations similar to that for finding symmetries. An explicit construction formula is derived which yields a conservation law for each solution of the determining system. In the first of two papers (Part I), examples of nonlinear wave equations are used to exhibit the method. Classification results for conservation laws of these equations are obtained. In a second paper (Part II), a general treatment of the method is given.

Harmonic maps, conservation laws, and moving frames

Abstract: Preface Introduction Acknowledgements Notations 1. Geometric and analytic setting 2. Harmonic maps with symmetries 3. Compensations and exotic function spaces 4. Harmonic maps without symmetries 5. Surfaces with mean curvature in L2 References.

Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves

Abstract: I. Fundamental concepts and examples.- 1. Hyperbolicity, genuine nonlinearity, and entropies.- 2. Shock formation and weak solutions.- 3. Singular limits and the entropy inequality.- 4. Examples of diffusive-dispersive models.- 5. Kinetic relations and traveling waves.- 1. Scalar Conservation Laws.- II. The Riemann problem.- 1. Entropy conditions.- 2. Classical Riemann solver.- 3. Entropy dissipation function.- 4. Nonclassical Riemann solver for concave-convex flux.- 5. Nonclassical Riemann solver for convex-concave flux.- III. Diffusive-dispersive traveling waves.- 1. Diffusive traveling waves.- 2. Kinetic functions for the cubic flux.- 3. Kinetic functions for general flux.- 4. Traveling waves for a given speed.- 5. Traveling waves for a given diffusion-dispersion ratio.- IV. Existence theory for the Cauchy problem.- 1. Classical entropy solutions for convex flux.- 2. Classical entropy solutions for general flux.- 3. Nonclassical entropy solutions.- 4. Refined estimates.- V. Continuous dependence of solutions.- 1. A class of linear hyperbolic equations.- 2. L1 continuous dependence estimate.- 3. Sharp version of the continuous dependence estimate.- 4. Generalizations.- 2. Systems of Conservation Laws.- VI. The Riemann problem.- 1. Shock and rarefaction waves.- 2. Classical Riemann solver.- 3. Entropy dissipation and wave sets.- 4. Kinetic relation and nonclassical Riemann solver.- VII. Classical entropy solutions of the Cauchy problem.- 1. Glimm interaction estimates.- 2. Existence theory.- 3. Uniform estimates.- 4. Pointwise regularity properties.- VIII. Nonclassical entropy solutions of the Cauchy problem.- 1. A generalized total variation functional.- 2. A generalized weighted interaction potential.- 3. Existence theory.- 4. Pointwise regularity properties.- IX. Continuous dependence of solutions.- 1. A class of linear hyperbolic systems.- 2. L1 continuous dependence estimate.- 3. Sharp version of the continuous dependence estimate.- 4. Generalizations.- X. Uniqueness of entropy solutions.- 1. Admissible entropy solutions.- 2. Tangency property.- 3. Uniqueness theory.- 4. Applications.

Direct construction method for conservation laws of partial differential equations. Part II: General treatment

Abstract: This paper gives a general treatment and proof of the direct conservation law method presented in Part I (see Anco & Bluman [3]). In particular, the treatment here applies to finding the local conservation laws of any system of one or more partial differential equations expressed in a standard Cauchy-Kovalevskaya form. A summary of the general method and its effective computational implementation is also given.

Central-upwind schemes for the saint-venant system

Abstract: We present one- and two-dimensional central-upwind schemes for approximating solutions of the Saint-Venant system with source terms due to bottom topography. The Saint-Venant system has steady-state solutions in which nonzero flux gradients are exactly balanced by the source terms. It is a challenging problem to preserve this delicate balance with numerical schemes. Small perturbations of these states are also very dicult to compute. Our approach is based on extending semi-discrete central schemes for systems of hyperbolic conservation laws to balance laws. Special attention is paid to the discretization of the source term such as to preserve stationary steady-state solutions. We also prove that the second-order version of our schemes preserves the nonnegativity of the height of the water. This important feature allows one to compute solutions for problems that include dry areas.

A Wave Propagation Method for Conservation Laws and Balance Laws with Spatially Varying Flux Functions

Abstract: We study a general approach to solving conservation laws of the form qt+f(q,x)x=0, where the flux function f(q,x) has explicit spatial variation. Finite-volume methods are used in which the flux is discretized spatially, giving a function fi(q) over the ith grid cell and leading to a generalized Riemann problem between neighboring grid cells. A high-resolution wave-propagation algorithm is defined in which waves are based directly on a decomposition of flux differences fi(Qi)-f-1(Qi-1) into eigenvectors of an approximate Jacobian matrix. This method is shown to be second-order accurate for smooth problems and allows the application of wave limiters to obtain sharp results on discontinuities. Balance laws $q_t+f(q,x)_x=\psi(q,x)$ are also considered, in which case the source term is used to modify the flux difference before performing the wave decomposition, and an additional term is derived that must also be included to obtain full accuracy. This method is particularly useful for quasi-steady problems close to steady state.

Almost Conservation Laws and Global Rough Solutions to a Nonlinear Schrödinger Equation

Abstract: We prove an “almost conservation law” to obtain global-in-time well-posedness for the cubic, defocussing nonlinear Schrodinger equation in Hs(R) when n = 2, 3 and s > 4 7 , 5 6 , respectively.

Global Weak Solutions for the Two-Dimensional Motion of Several Rigid Bodies in an Incompressible Viscous Fluid

Abstract: We consider the two-dimensional motion of several non-homogeneous rigid bodies immersed in an incompressible non-homogeneous viscous fluid. The fluid, and the rigid bodies are contained in a fixed open bounded set of R 2 . The motion of the fluid is governed by the Navier-Stokes equations for incompressible fluids and the standard conservation laws of linear and angular momentum rule the dynamics of the rigid bodies. The time variation of the fluid domain (due to the motion of the rigid bodies) is not known a priori, so we deal with a free boundary value problem. The main novelty here is the demonstration of the global existence of weak solutions for this problem. More precisely, the global character of the solutions we obtain is due to the fact that we do not need any assumption concerning the lack of collisions between several rigid bodies or between a rigid body and the boundary. We give estimates of the velocity of the bodies when their mutual distance or the distance to the boundary tends to zero.

An objective finite element approximation of the kinematics of geometrically exact rods and its use in the formulation of an energy-momentum conserving scheme in dynamics

Abstract: We present in this paper a new finite element formulation of geometrically exact rod models in the three-dimensional dynamic elastic range. The proposed formulation leads to an objective (or frame-indifferent under superposed rigid body motions) approximation of the strain measures of the rod involving finite rotations of the director frame, in contrast with some existing formulations. This goal is accomplished through a direct finite element interpolation of the director fields defining the motion of the rod's cross-section. Furthermore, the proposed framework allows the development of time-stepping algorithms that preserve the conservation laws of the underlying continuum Hamiltonian system. The conservation laws of linear and angular momenta are inherited by construction, leading to an improved approximation of the rod's dynamics. Several numerical simulations are presented illustrating these properties. Copyright © 2002 John Wiley & Sons, Ltd.

Stresses in lipid membranes

Abstract: The stresses in a closed lipid membrane described by the Helfrich Hamiltonian, quadratic in th ee xtrinsic curvature, are identified using Noether’s theorem. Three equations describe the conservation of the stress tensor: the norma lp rojection is identified as the shape equation describing equilibrium configurations; the tangential projections are consistency conditions on the stresses which capture the fluid character of such membranes. The corresponding torque tensor is also identified. The use of the stress tensor as a basis for perturbation theory is discussed. The conservation laws are cast in terms of the forces and torques on closed curves. As an application, the first integral of the shape equation for axially symmetric configurations is derived by examining the forces which are balanced along the circles of constant latitude.

Adaptive Discontinuous Galerkin Finite Element Methods for Nonlinear Hyperbolic Conservation Laws

Abstract: We consider the a posteriori error analysis and adaptive mesh design for discontinuous Galerkin finite element approximations to systems of nonlinear hyperbolic conservation laws. In particular, we discuss the question of error estimation for general target functionals of the solution; typical examples include the outflow flux, local average and pointwise value, as well as the lift and drag coefficients of a body immersed in an inviscid fluid. By employing a duality argument, we derive so-called weighted or Type I a posteriori error bounds; these error estimates include the product of the finite element residuals with local weighting terms involving the solution of a certain dual or adjoint problem that must be numerically approximated. Based on the resulting approximate Type I bound, we design and implement an adaptive algorithm that produces meshes specifically tailored to the efficient computation of the given target functional of practical interest. The performance of the proposed adaptive strategy and the quality of the approximate Type I a posteriori error bound is illustrated by a series of numerical experiments. In particular, we demonstrate the superiority of this approach over standard mesh refinement algorithms which employ Type II error indicators.

Fully Discrete, Entropy Conservative Schemes of Arbitrary Order

Abstract: 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].

A comparison of four approaches to the calculation of conservation laws

Abstract: The paper compares computational aspects of four approaches to compute conservation laws of single Differential Equations (DEs) or systems of them, ODEs and PDEs. The only restriction, required by two of the four corresponding computer algebra programs, is that each DE has to be solvable for a leading derivative. Extra constraints for the conservation laws can be specified. Examples include new conservation laws that are non-polynomial in the functions, that have an explicit variable dependence and families of conservation laws involving arbitrary functions. The following equations are investigated in examples: Ito, Liouville, Burgers, Kadomtsev–Petviashvili, Karney–Sen–Chu–Verheest, Boussinesq, Tzetzeica, Benney.

A Fourth-Order Central WENO Scheme for Multidimensional Hyperbolic Systems of Conservation Laws

Abstract: We present the first fourth-order central scheme for two-dimensional hyperbolic systems of conservation laws. Our new method is based on a central weighted nonoscillatory approach. The heart of our method is the reconstruction step, in which a genuinely two-dimensional interpolant is reconstructed from cell averages by taking a convex combination of building blocks in the form of biquadratic polynomials. Similarly to other central schemes, our new method enjoys the simplicity of the black-box approach. All that is required in order to solve a problem is to supply the flux function and an estimate on the speed of propagation. The high-resolution properties of the scheme as well as its resistance to mesh orientation, and the effectiveness of the componentwise approach, are demonstrated in a variety of numerical examples.

Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation

Abstract: The generalized Korteweg-de Vries equations are a class of Hamiltonian systems in infinite dimension derived from the KdV equation where the quadratic term is replaced by a higher order power term. These equations have two conservation laws in the energy space H 1 (L 2 norm and energy). We consider in this paper the critical generalized KdV equation, which corresponds to the smallest power of the nonlinearity such that the two conservation laws do not imply a bound in H 1 uniform in time for all H 1 solutions (and thus global existence). From [15], there do exist for this equation solutions u(t) such that |u(t)| H1 → +∞ as t ↑ T, where T < +∞ (we call them blow-up solutions). The question is to describe, in a qualitative way, how blow up occurs. For solutions with L 2 mass close to the minimal mass allowing blow up and with decay in L 2 at the right, we prove after rescaling and translation which leave invariant the L 2 norm that the solution converges to a universal profile locally in space at the blow-up time T. From the nature of this profile, we improve the standard lower bound on the blow-up rate for finite time blow-up solutions.

A Basis of Conservation Laws for Partial Differential Equations

Abstract: The classical generation theorem of conservation laws from known ones for a system of differential equations which uses the action of a canonical Lie–Backlund generator is extended to include any L...

Geometric Integrability of the Camassa–Holm Equation

Abstract: It is observed that the Camassa–Holm equation describes pseudo-spherical surfaces and that therefore, its integrability properties can be studied by geometrical means. An sl(2, R)-valued linear problem whose integrability condition is the Camassa–Holm equation is presented, a ‘Miura transform’ and a ‘modified Camassa–Holm equation’ are introduced, and conservation laws for the Camassa–Holm equation are then directly constructed. Finally, it is pointed out that this equation possesses a nonlocal symmetry, and its flow is explicitly computed.

Viscoelastic flows studied by smoothed particle dynamics

Abstract: A viscoelastic numerical scheme based on smoothed particle dynamics is presented. The concept goes a step beyond smoothed particle hydrodynamics (SPH) which is a grid-free Lagrangian method describing the flow by fluid-pseudo-particles. The relevant properties are interpolated directly on the resulting movable grid. In this work, the effect of viscoelasticity is incorporated into the ordinary conservation laws by a differential constitutive equation supplied for the stress tensor. In order to give confidence in the methodology we explicitly consider the non-stationary simple corotational Maxwell model in a channel geometry. Without further developments the scheme is applicable to ‘realistic’ models relevant for three-dimensional (3D) viscoelastic flows in complex geometries.

On the Artificial Compression Method for Second-Order Nonoscillatory Central Difference Schemes for Systems of Conservation Laws

Abstract: The recently proposed high-order central difference schemes for conservation laws have a tendency of smearing linear discontinuities. In principle, Harten's artificial compression method (ACM) could be used to improve resolution. We analyze why this approach has not yet been used successfully and derive a more powerful version of the ACM based on a rigorous estimate of the total variation. We discuss the potential danger of overcompression and point out directions of future algorithmic development.

On the Properties of Time-Dependent, Force-Free, Degenerate Electrodynamics

Abstract: This paper formulates time-dependent, force-free, degenerate electrodynamics as a hyperbolic system of conservation laws. It is shown that this system has four characteristic modes, a pair of fast waves propagating with the speed of light and a pair of Alfven waves. All these modes are linearly degenerate. The results of this analytic study can be used in developing upwind numerical schemes for the electrodynamics of black holes and pulsar magnetospheres. As an example, this paper describes a simple one-dimensional numerical scheme based on linear and exact Riemann solvers.