Duality for systems of conservation laws

Letters in Mathematical Physics - For one-dimensional systems of conservation laws admitting two additional conservation laws, we assign a ruled hypersurface of codimension two in projective space....     

Positivity-preserving high-resolution schemes for systems of conservation laws

A new class of flux-limited schemes for systems of conservation laws is presented that is both high-resolution and positivity-preserving. The schemes are obtained by extending the Steger–Warming method to second-order accuracy through the use of component-wise TVD flux limiters while ensuring that the coefficients of the discretization equation are positive. A coefficient is considered positive if it has all-positive eigenvalues and has the same eigenvectors as those of the convective flux Jacobian evaluated at the corresponding node. For certain systems of conservation laws, such as the Euler equations for instance, this condition is sufficient to guarantee positivity-preservation. The method proposed is advantaged over previous positivity-preserving flux-limited schemes by being capable to capture with high resolution all wave types (including contact discontinuities, shocks, and expansion fans). Several test cases are considered in which the Euler equations in generalized curvilinear coordinates are solved in 1D, 2D, and 3D. The test cases confirm that the proposed schemes are positivity-preserving while not being significantly more dissipative than the conventional TVD methods. The schemes are written in general matrix form and can be used to solve other systems of conservation laws, as long as they are homogeneous of degree one.     

Symmetries and conservation laws for generalized hamiltonian systems

A class of dynamical systems which locally correspond to a general first-order system of Euler-Lagrange equations is studied on a contact manifold. These systems, called self-adjoint, can be regarded as generalizations of (time-dependent) Hamiltonian systems. It is shown that each one-parameter family of symmetries of the underlying contact form defines a parameter-dependent constant of the motion and vice versa. Next, an extension of the classical concept of canonical transformations is introduced. One-parameter families of canonical transformations are studied and shown to be generated as solutions of a self-adjoint system. Some of the results are illustrated on the Emden equation.     

Pseudo-spectral methods for the spatial symplectic reduction of open systems of conservation laws

A reduction method is presented for systems of conservation laws with boundary energy flow. It is stated as a generalized pseudo-spectral method which performs exact differentiation by using simultaneously several approximation spaces generated by polynomials bases and suitable choices of port-variables. The symplecticity of this spatial reduction method is proved when used for the reduction of both closed and open systems of conservation laws, for any choice of collocation points (i.e. for any polynomial bases). The symplecticity of some more usual collocation schemes is discussed and finally their accuracy on approximation of the spectrum, on the example of the ideal transmission line, is discussed in comparison with the suggested reduction scheme.     

Multi-level Monte Carlo finite volume methods for nonlinear systems of conservation laws in multi-dimensions

We extend the multi-level Monte Carlo (MLMC) in order to quantify uncertainty in the solutions of multi-dimensional hyperbolic systems of conservation laws with uncertain initial data. The algorithm is presented and several issues arising in the massively parallel numerical implementation are addressed. In particular, we present a novel load balancing procedure that ensures scalability of the MLMC algorithm on massively parallel hardware. A new code is described and applied to simulate uncertain solutions of the Euler equations and ideal magnetohydrodynamics (MHD) equations. Numerical experiments showing the robustness, efficiency and scalability of the proposed algorithm are presented.     

Multi-moment ADER-Taylor methods for systems of conservation laws with source terms in one dimension

A new integration method combining the ADER time discretization with a multi-moment finite-volume framework is introduced. ADER runtime is reduced by performing only one Cauchy–Kowalewski (C–K) procedure per cell per time step and by using the Differential Transform Method for high-order derivatives. Three methods are implemented: (1) single-moment WENO (WENO), (2) two-moment Hermite WENO (HWENO), and (3) entirely local multi-moment (MM-Loc). MM-Loc evolves all moments, sharing the locality of Galerkin methods yet with a constant time step during p-refinement.Five 1-D experiments validate the methods: (1) linear advection, (2) Burger’s equation shock, (3) transient shallow-water (SW), (4) steady-state SW simulation, and (5) SW shock. WENO and HWENO methods showed expected polynomial h-refinement convergence and successfully limited oscillations for shock experiments. MM-Loc showed expected polynomial h-refinement and exponential p-refinement convergence for linear advection and showed sub-exponential (yet super-polynomial) convergence with p-refinement in the SW case.HWENO accuracy was generally equal to or better than a five-moment MM-Loc scheme. MM-Loc was less accurate than RKDG at lower refinements, but with greater h- and p-convergence, RKDG accuracy is eventually surpassed. The ADER time integrator of MM-Loc also proved more accurate with p-refinement at a CFL of unity than a semi-discrete RK analog of MM-Loc. Being faster in serial and requiring less frequent inter-node communication than Galerkin methods, the ADER-based MM-Loc and HWENO schemes can be spatially refined and have the same runtime, making them a competitive option for further investigation.     

Torsion and conservation laws

The interaction of spinor and electromagnetic fields with the torsion of space-time is studied within the framework of the Einstein-Cartan theory. The equivalent nonlinear theory in Riemann space is obtained. The conservation laws for the vector and pseudovector currents are investigated in the nonlinear theory.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 50–52, February, 1986.     

Syzygies and conservation laws

We show how the notion of syzygies of a system of partial differential equations allows to derive some conservation laws, for the case of Maxwell and Proca systems. More in general, we apply some classical tools in algebraic analysis to derive properties of the solutions of the previous systems like their integral representations.     

The conservation laws for deformed classical models

The problem of deriving the conservation laws for deformed linear equations of motion is investigated. The conserved currents are obtained in the explicit form and used in the construction of constants of motion. The equations for the set of non-interacting oscillators with arbitrary scale-time as well as theκ-Klein-Gordon equation are considered as an example of application of the method.     

Entropy viscosity method for nonlinear conservation laws

A new class of high-order numerical methods for approximating nonlinear conservation laws is described (entropy viscosity method). The novelty is that a nonlinear viscosity based on the local size of an entropy production is added to the numerical discretization at hand. This new approach does not use any flux or slope limiters, applies to equations or systems supplemented with one or more entropy inequalities and does not depend on the mesh type and polynomial approximation. Various benchmark problems are solved with finite elements, spectral elements and Fourier series to illustrate the capability of the proposed method.     

Chiral anomaly in Weyl systems: No violation of classical conservation laws

The anomalous term $～\mathbf{E}\mathbf{B}$ in the balance of the chiral density can be rewritten as quantum current in the classical balance of density. Therefore it does not violate conservation laws as sometimes claimed to be caused by quantum fluctuations.     

Hyperbolic regularizations of conservation laws

One of the problems of the kinetics of nonequilibrium processes is related to the lack of information concerning most of the nonequilibrium variables, namely, those which have no intuitive physical meaning, i.e., cannot be defined from the experiment. Moreover, the number of nonequilibrium variables is so large that a reasonable amount (from the physical point of view) of boundary conditions is insufficient for posing the mixed problem. What do the initial data for the Cauchy problem and the boundary conditions for the mixed problem mean in this case? In fact, we must assume that the initial-boundary data for most of the nonequilibrium variables (the higher-order momenta) are arbitrary! The British physicists Chapman and Enskog conjectured that, for “physically correct” models of continuum mechanics, the influence of the higher-order momenta is “inessential.” There are some postulates of physical correctness, but we do not dwell on them. For us it is of importance to understand what the fact that the influence of the higher-order momenta is “inessential” means from the mathematical point of view. The paper is devoted to this very topic.     

Long-time effect of relaxation for hyperbolic conservation laws

In processes such as invasion percolation and certain models of continuum percolation, in which a possibly random labelf(b) is attached to each bondb of a possibly random graph, percolation models for various values of a parameterr are naturally coupled: one can define a bondb to be occupied at levelr iff(b)r. If the labeled graph is stationary, then under the mild additional assumption of positive finite energy, a result of Gandolfi, Keane, and Newman ensures that, in lattice models, for each fixedr at which percolation occurs, the infinite cluster is unique a.s. Analogous results exist for certain continuum models. A unifying framework is given for such fixed-r results, and it is shown that if the site density is finite and the labeled graph has positive finite energy, then with probability one, uniqueness holds simultaneously for all values ofr. An example is given to show that when the site density is infinite, positive finite energy does not ensure uniqueness, even for fixedr. In addition, with finite site density but without positive finite energy, one can have fixed-r uniqueness a.s. for eachr, yet not have simultaneous uniqueness.Research supported by NSF grant DMS-9206139     

Noether equations and conservation laws

The purpose of the paper is to present a rigorous derivation of the relation between conservation laws and transformations leaving invariant the action integral. The (space-)time development of a physical system is represented by a cross section of a product bundleM. A Lagrange function is defined as a mapping where is the bundle space of the first jet extension ofM. A symmetry transformation is defined as a bundle automorphism ofM, carrying solutions of the Euler-Lagrange equation into solutions of the same equation. An important class of symmetry transformations is that of generalized invariant transformations: they are defined by specifying their action on the Euler-Lagrange equation. The generators of generalized invariant transformations are solutions of a system of linear, homogeneous partial differential equation (Noether equations). The set of all solutions of these equations has a natural structure of Lie algebra. In a simple manner, the Noether equations give rise to differential conservation laws.Supported by Air Force Office of Scientifie Research and Aeronautical Research Laboratories.On leave of absence from the Institute of Theoretical Physics, Warsaw University, Warsaw, Poland.     

Worldtube conservation laws for the null-timelike evolution problem

I treat the worldtube constraints which arise in the null-timelike initial-boundary value problem for the Bondi-Sachs formulation of Einstein’s equations. Boundary data on a worldtube and initial data on an outgoing null hypersurface determine the exterior spacetime by integration along the outgoing null geodsics. The worldtube constraints are a set of conservation laws which impose conditions on the integration constants. I show how these constraints lead to a well-posed initial value problem governing the extrinsic curvature of the worldtube, whose components are related to the integration constants. Possible applications to gravitational waveform extraction and to the well-posedness of the null-timelike initial-boundary value problem are discussed.     

Energy conservation laws and antimatter

In field theory, an energy-momentum tensor fails to be conserved if internal symmetries are broken. This revised version was published online in August 2006 with corrections to the Cover Date.     

Hyperbolic conservation laws with relaxation

The effect of relaxation is important in many physical situations. It is present in the kinetic theory of gases, elasticity with memory, gas flow with thermo-non-equilibrium, water waves, etc. The governing equations often take the form of hyperbolic conservation laws with lower-order terms. In this article, we present and analyze a simple model of hyperbolic conservation laws with relaxation effects. Dynamic subcharacteristics governing the propagation of disturbances over strong wave forms are identified. Stability criteria for diffusion waves, expansion waves and traveling waves are found and justified nonlinearly. Time-asymptotic expansion and the energy method are used in the analysis. For dissipative waves, the expansion is similar in spirit to the Chapman-Enskog expansion in the kinetic theory. For shock waves, however, a different approach is needed.The paper was written at Mittag-Leffler Institute; the author wants to thank the Institute for the visiting position in 1986. This work was supported in part by an NSF grant