One-sided stability and convergence of the Nessyahu–Tadmor scheme

Non-oscillatory schemes are widely used in numerical approximations of nonlinear conservation laws. The Nessyahu–Tadmor (NT) scheme is an example of a second order scheme that is both robust and simple. In this paper, we prove a new stability property of the NT scheme based on the standard minmod reconstruction in the case of a scalar strictly convex conservation law. This property is similar to the One-sided Lipschitz condition for first order schemes. Using this new stability, we derive the convergence of the NT scheme to the exact entropy solution without imposing any nonhomogeneous limitations on the method. We also derive an error estimate for monotone initial data.     

Rate Conservation Laws for Multidimensional Processes of Bounded Variation with Applications to Priority Queueing Systems

In this paper we derive a multidimensional version of the rate conservation law (RCL) for càdlàg processes of bounded variation. These results are then used to analyze queueing models which have a natural multidimensional characterization, such as priority queues. In particular the RCL is used to establish certain conservation laws between the idle probabilities for such queues. We use the relations to provide a detailed analysis of preemptive resume priority queues with M/G inputs. Special attention is paid to the validity of the so-called reduced service rate approximation.     

Analyse d'un schéma décalé pour l'approximation de systèmes hyperboliques non conservatifs

We consider a model system made of two nonlinear equations which are non conservative. A conservation law can be obtained from these equations through linear operations only, which don't modify the shock waves. A numerical scheme based on a different mesh adapted to each variable is proposed. By choosing a shifted mesh, we have un explicit Riemann solver and we can derive a finite volume scheme. We prove a priori estimates in L^∞ norm and Total Variation for the system, which lead to a strong convergence in L¹ norm towards a solution satisfying the associated conservation law.     

Conservation laws for Lotka–Volterra models

We derive necessary and sufficient conditions on a Lotka–Volterra model to admit a conservation law of Volterra's type. The result and the proof for the corresponding linear algebra problem are given in graph‐theoretical terms; they refer to the directed graph which is defined by the coefficients of the differential equation system. Copyright © 2003 John Wiley & Sons, Ltd.     

On a hyperbolic conservation law of electron transport in solid materials for electron probe microanalysis

The prediction of X-ray intensities based on the distribution of electrons throughout solid materials is essential to solve the inverse problem of quantifying the composition of materials in electron probe microanalysis (EPMA) [3]. We present a hyperbolic conservation law for electron transport in solid materials and investigate its validity under conditions typical for EPMA experiments. The conservation law is based on the time-stationary Boltzmann equation for binary electron-atom scattering. We model the energy loss of the electrons with a continuous slowing-down approximation. A first order moment approximation with respect to the angular variable is discussed. We propose to use a minimum entropy closure to derive a system of hyperbolic conservation laws, known as the M1 model [11]. A finite volume scheme for the numerical solution of the resulting equations is presented. Important numerical aspects of the scheme are discussed, such as bounds for the finite propagation speeds, as well as difficulties arising fromspatial discontinuities in thematerial coefficients and the scaling of the characteristic velocities with the stopping power of the electrons.We compare the accuracy and performance of the numerical solution of the hyperbolic conservation law to Monte Carlo simulations. The results indicate a reasonable accuracy of the proposed method and showthat compared to the MonteCarlo simulation the finite volume scheme is computationally less expensive.     

Conservation laws for Camassa–Holm equation, Dullin–Gottwald–Holm equation and generalized Dullin–Gottwald–Holm equation

In this paper we construct the conservation laws for the Camassa–Holm equation, the Dullin–Gottwald–Holm equation (DGH) and the generalized Dullin–Gottwald–Holm equation (generalized DGH). The variational derivative approach is used to derive the conservation laws. Only first order multipliers are considered. Two multipliers are obtained for the Camassa–Holm equation. For the DGH and generalized DGH equations the variational derivative approach yields two multipliers; thus two conserved vectors are obtained.     

On the Boundary Condition at the Surface of a Porous Medium

A theoretical justification is given for an empirical boundary condition proposed by Beavers and Joseph [1]. The method consists of first using a statistical approach to extend Darcy's law to non homogeneous porous medium. The limiting case of a step function distribution of permeability and porosity is then examined by boundary layer techniques, and shown to give the required boundary condition. In an Appendix, the statistical approach is checked by using it to derive Einstein's law for the viscosity of dilute suspensions.     

Noether-Type Discrete Conserved Quantities Arising from a Finite Element Approximation of a Variational Problem

In this work, we prove a weak Noether-type Theorem for a class of variational problems that admit broken extremals. We use this result to prove discrete Noether-type conservation laws for a conforming finite element discretisation of a model elliptic problem. In addition, we study how well the finite element scheme satisfies the continuous conservation laws arising from the application of Noether’s first theorem (1918). We summarise extensive numerical tests, illustrating the conservation of the discrete Noether law using the p-Laplacian as an example and derive a geometric-based adaptive algorithm where an appropriate Noether quantity is the goal functional.     

Local Conservation Laws and the Hamiltonian Formalism for the Ablowitz–Ladik Hierarchy

We derive a systematic and recursive approach to local conservation laws and the Hamiltonian formalism for the Ablowitz–Ladik (AL) hierarchy. Our methods rely on a recursive approach to the AL hierarchy using Laurent polynomials and on asymptotic expansions of the Green's function of the AL Lax operator, a five-diagonal finite difference operator.     

Liouville-type results for stationary maps of a class of functional related to pullback metrics

We study a generalized functional related to the pullback metrics (3). We derive the first variation formula which yield stationary maps. We introduce the stress–energy tensor which is naturally linked to conservation law and yield the monotonicity formula via the coarea formula and the comparison theorem in Riemannian geometry. A version of this monotonicity inequalities enables us to derive some Liouville type results. Also, we investigate the constant Dirichlet boundary value problems and the generalized Chern type results for tension field equation with respect to this functional.     

Energy flows of a gravitational wave in a liquid sphere

We consider the energy transfer problem in a static space-time background created by a liquid spherical body. In the accompanying reference frame, we derive the Lagrangian of gravity and the liquid that is of the fourth order in perturbations. We impose gauge conditions and integrate over the angular coordinates at the level of the action, which makes the problem two-dimensional. We derive the density and flow of the perturbation energy. Different gauge choices are considered. The energy conservation law is ensured by the static property of the metric and by the vanishing of the Lagrange variations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 120, No. 2, pp. 342–351, August, 1999.     

A characterization of the blow-up time for the solution of a conservation law in several space variable

We derive a necessary and sufficient condition on the L^∞ Cauchy data for a conservation law in several space variables under which the solution will be locally Lipschitz continuous up to time T . The largesf such T is therefore the “blow-up” time. Roughly, our condition is that the data can be approximated by smoother functions satisfying uniformly a certain estimate. We present an example which shows that the existence of the approximations is crucial: it is not sufficient that the data itself satisfy this estimate.     

Conservation laws for a complexly coupled KdV system,coupled Burgers’ system and Drinfeld–Sokolov–Wilson system via multiplier approach

The multiplier approach (variational derivative method) is used to derive the conservation laws for some nonlinear systems of partial differential equations. Firstly, the multipliers (characteristics) are computed and then conserved vectors are obtained for the each multiplier. Examples of the third-order complexly coupled KdV system, second-order coupled Burgers’ system and third-order Drinfeld–Sokolov–Wilson system are considered. For all three systems the local conservation laws are established by utilizing the multiplier approach.     

Conservation laws for laminar axisymmetric jet flows with weak swirl

The conservation laws for laminar axisymmetric jet flows with weak swirl are studied here. The multiplier approach is used to derive the conservation laws for the system of three boundary layer equations for the velocity components governing flow in laminar axisymmetric jet flows with weak swirl. Conservation laws for the system of two partial differential equations for the stream function are also derived.     

On integrals of equations of unstable nearly self-similar flows : PMM vol. 39, n 6, 1975, pp. 1060–1067

One-dimensional or nearly one-dimensional unstable motions of perfect gas are considered. Integrals admitted by the system of equations defining such motions are examined. Since the existence of integrals is associated with some law of conservation, i. e. with some divergent form of presentation of equations of the input system, it is possible by examining all divergent equations of gasdynamics to derive certain new integrals not previously considered.     

Criterion of hyperbolicity for non‐conservative quasilinear systems admitting a partially convex conservation law

A system of conservation laws admitting an additional convex conservation law can be written as a symmetric t‐hyperbolic in the sense of Friedrichs system. However, in mathematical modeling of complex physical phenomena, it is customary to use non‐conservative hyperbolic models. We generalize the Godunov–Friedrichs–Lax approach to this new class of models. Copyright © 2011 John Wiley & Sons, Ltd.     

Well-posedness and blow-up for a modified two-component Camassa–Holm equation

In this article, we consider a newly modified two-component Camassa–Holm equation. First, we establish the local well-posedness result, then we present a precise blow-up scenario. Afterwards, we derive a new conservation law, by which and the precise blow-up scenario we prove three blow-up results and a blow-up rate estimate result.     

Existence of a Global Solution for a Scalar Conservation Law with a Source Term

We are concerned with a scalar conservation law with a source term. This equation is proposed to describe the qualitative behavior of waves for a general system in resonance with the source term by T.P. Liu. In addition to this, the scalar conservation law is used in various areas such as fluid dynamics, traffic problems etc.In the present paper, we prove the global existence and stability of entropy solutions to the Cauchy problem. The difficult point is to obtain the bounded estimate of solutions. To solve it, we introduce some functions as the lower and upper bounds. Therefore, our bounded estimate depends on the space variable. This idea comes from the generalized invariant region theory for the compressible Euler equation. The method is also applicable to other nonlinear problems involving similar difficulties. Finally, we use the vanishing viscosity method to construct approximate solutions and derive the convergence by the compensated compactness.     

重建极性连续统理论的基本定律和原理(Ⅱ)——微态连续统理论和偶应力理论

重建微态连续统理论和偶应力理论的动量和动量矩均衡定律以及能量守恒定律,并由这些定律自然地推导出相应的局部和非局部均衡方程。这些结果可由耦合型微极连续统理论过渡和归结而得到。把推导出的结果和传统的质量和微惯性守恒定律以及熵不等式结合在一起就构成微态连续统理论和偶应力理论的基本均衡定律和方程体系。还弄清了以前的各种连续统理论的不完整性层次。最后,给出了几种特殊情形。     

Equation of motion for a spherically-symmetric shell in the relativistic theory of gravity

In the framework of the relativistic theory of gravity, the equation of motion for a spherically-symmetric singular shell is derived and integrated in the first approximation of the Newton potential U = m/r. We use the covariant energy-momentum conservation law for matter in the effective Riemannian space and, independently, the energy-momentum conservation law for the matter + gravity system in Minkowski space. For the problem under consideration, we show the equivalence of our approach to the classical formalism of singular shells.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 2, pp. 344–352, May, 1996.Translated by A. M. Semikhatov.