Potential Symmetries and Associated Conservation Laws with Application to Wave Equations

It has been shown that one can generate a class of nontrivial conservation laws for second-order partial differential equations using some recent results dealing with the action of any Lie–Bäcklund symmetry generator of the equivalentfirst-order system on the respective conservation law. These conservedvectors are nonlocal as they are constructed from associatednonlocal symmetries of the partial differential equation. The method canbe successfully extended to association with genuine nonlocal(potential) symmetries. However, it usually involves solving moredifficult systems of partial differential equations which may not alwaysbe easy to uncouple.     

Noether-Type Symmetries and Conservation Laws Via Partial Lagrangians

We show how one can construct conservation laws of Euler-Lagrange-type equations via Noether-type symmetry operators associated with what we term partial Lagrangians. This is even in the case when a system does not directly have a usual Lagrangian, e.g. scalar evolution equations. These Noether-type symmetry operators do not form a Lie algebra in general. We specify the conditions under which they do form an algebra. Furthermore, the conditions under which they are symmetries of the Euler-Lagrange-type equations are derived. Examples are given including those that admit a standard Lagrangian such as the Maxwellian tail equation, and equations that do not such as the heat and nonlinear heat equations. We also obtain new conservation laws from Noether-type symmetry operators for a class of nonlinear heat equations in more than two independent variables.     

On Material Equations in Second Gradient Electroelasticity

The concern of this work is the derivation of material conservation and balance laws for second gradient electroelasticity. The conservation laws of material momentum, material angular momentum and scalar moment of momentum on the material manifold are derived using Noether's theorem and the exact conditions under which they hold are rigorously studied. The corresponding balance laws are also presented. This revised version was published online in June 2006 with corrections to the Cover Date.     

On Conservation and Balance Laws in Micromorphic Elastodynamics

In this paper, following Noether’s theorem we investigate the Lie point symmetries of linear micromorphic elastodynamics (linear elastodynamics with microstructure). Conservation and balance laws of linear, micromorphic elastodynamics are derived. We generalize the J, L and M integrals for this theory. In addition, we give the Eshelby stress tensor, pseudomomentum vector, field intensity vector, Hamiltonian, angular momentum tensor and scaling flux generalized to micromorphic elastodynamics.      

Lie–Bäcklund and Noether Symmetries with Applications

New identities relating the Euler–Lagrange, Lie–Bäcklund and Noether operators are obtained. Some important results are shown to be consequences of these fundamental identities. Furthermore, we generalise an interesting example presented by Noether in her celebrated paper and prove that any Noether symmetry is equivalent to a strict Noether symmetry, i.e. a Noether symmetry with zero divergence. We then use the symmetry based results deduced from the new identities to construct Lagrangians for partial differential equations. In particular, we show how the knowledge of a symmetry and its corresponding conservation law of a given partial differential equation can be utilised to construct a Lagrangian for the equation. Several examples are given.     

The Connection Between Isometries and Symmetries of Geodesic Equations of the Underlying Spaces

A connection between the symmetries of manifolds and differential equations is sought through the geodesic equations of maximally symmetric spaces, which have zero, constant positive or constant negative curvature. It is proved that for a space admitting so(n+1) or so(n,1) as the maximal isometry algebra, the symmetry of the geodesic equations of the space is given by so( or (where d ₂ is the two-dimensional dilation algebra), while for those admitting (where represents semidirect sum) the algebra is sl(n+2). A corresponding result holds on replacing so(n) by so(p,q) with p+q = n. It is conjectured that if the isometry algebra of any underlying space of non-zero curvature is h, then the Lie symmetry algebra of the geodesic equations is given by , provided that there is no cross-section of zero curvature at the point under consideration. If there is a flat subspace of dimension m, then the symmetry group becomes ).     

Lie Point Symmetries,Conservation and Balance Laws in Linear Gradient Elastodynamics

The aim of this work is the derivation of Lie point symmetries, conservation and balance laws in linear gradient elastodynamics of grade-2 (up to second gradients of the displacement vector and the first gradient of the velocity). The conservation and balance laws of translational, rotational, scaling variational symmetries and addition of solutions are derived using Noether’s theorem. It turns out that the scaling symmetry is not a strict variational symmetry in gradient elasticity.      

New Conservation Laws of Energy and C-D Inequalities in Continua Without Microstructure

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New Conservation Laws of Energy and C-D Inequalities in Continua with Microstructure

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On Dual Conservation Laws in Linear Elasticity: Stress Function Formalism

Dual conservation laws of linear planar elasticity theory have been systematically studied based on stress function formalism. By employing generalized symmetry transformation or the Lie—Bäcklund transformation, a class of new dual conservation laws in planar elasticity have been discovered based on the Noether theorem and its Bessel—Hagen generalization. The physical implications of these dual conservation laws are discussed briefly.     

Inverse Approach in Ordinary Differential Equations: Applications to Lagrangian and Hamiltonian Mechanics

This paper is on the so called inverse problem of ordinary differential equations, i.e. the problem of determining the differential system satisfying a set of given properties. More precisely we characterize under very general assumptions the ordinary differential equations in \(\mathbb {R}^N\) which have a given set of either \(M\) partial integrals, or \(M first integral, or \(M partial and first integrals. Moreover, for such systems we determine the necessary and sufficient conditions for the existence of \(N-1\) independent first integrals. We give two relevant applications of the solutions of these inverse problem to constrained Lagrangian and Hamiltonian systems respectively. Additionally we provide the general solution of the inverse problem in dynamics.     

Degenerate Hyperbolic Conservation Laws with Dissipation: Reduction to and Validity of a Class of Burgers-Type Equations

A conservation law is said to be degenerate or critical if the Jacobian of the flux vector evaluated on a constant state has a zero eigenvalue. In this paper, it is proved that a degenerate conservation law with dissipation will generate dynamics on a long time scale that resembles Burger’s dynamics. The case of k-fold degeneracy is also treated, and it is shown that it leads to a reduction to a quadratically coupled k-fold system of Burgers-type equations. Validity of the reduction and existence for the reduced system is proved in the class of uniformly local spaces, thereby capturing both finite and infinite energy solutions. The theory is applied to some examples, from stratified shallow-water hydrodynamics, that model the birth of hydraulic jumps.     

Flux and source term discretization in two‐dimensional shallow water models with porosity on unstructured grids

Two‐dimensional shallow water models with porosity appear as an interesting path for the large‐scale modelling of floodplains with urbanized areas. The porosity accounts for the reduction in storage and in the exchange sections due to the presence of buildings and other structures in the floodplain. The introduction of a porosity into the two‐dimensional shallow water equations leads to modified expressions for the fluxes and source terms. An extra source term appears in the momentum equation. This paper presents a discretization of the modified fluxes using a modified HLL Riemann solver on unstructured grids. The source term arising from the gradients in the topography and in the porosity is treated in an upwind fashion so as to enhance the stability of the solution. The Riemann solver is tested against new analytical solutions with variable porosity. A new formulation is proposed for the macroscopic head loss in urban areas. An application example is presented, where the large scale model with porosity is compared to a refined flow model containing obstacles that represent a schematic urban area. The quality of the results illustrates the potential usefulness of porosity‐based shallow water models for large scale floodplain simulations. Copyright © 2005 John Wiley & Sons, Ltd.     

3-D kinematical conservation laws (KCL): Evolution of a surface in – in particular propagation of a nonlinear wavefront

3-D KCL are equations of evolution of a propagating surface (or a wavefront) Ω_t in 3-space dimensions and were first derived by Giles, Prasad and Ravindran in 1995 assuming the motion of the surface to be isotropic. Here we discuss various properties of these 3-D KCL. These are the most general equations in conservation form, governing the evolution of Ω_t with singularities which we call kinks and which are curves across which the normal n to Ω_t and amplitude w on Ω_t are discontinuous. From KCL we derive a system of six differential equations and show that the KCL system is equivalent to the ray equations of Ω_t. The six independent equations and an energy transport equation (for small amplitude waves in a polytropic gas) involving an amplitude w (which is related to the normal velocity m of Ω_t) form a completely determined system of seven equations. We have determined eigenvalues of the system by a very novel method and find that the system has two distinct nonzero eigenvalues and five zero eigenvalues and the dimension of the eigenspace associated with the multiple eigenvalue 0 is only 4. For an appropriately defined m, the two nonzero eigenvalues are real when m>1 and pure imaginary when m<1. Finally we give some examples of evolution of weakly nonlinear wavefronts.     

格子玻尔兹曼方法概述：详尽的历史性文献及待探讨的理论问题

本文给出了详尽的格子玻尔兹曼方法的概述：包含其理论基础、起源、基本思想及主要特征,历史进展、重要的综述、著名书刊、应用及可用计算机代码,从而为从事有关研究的学生与研究人员提供了丰富的参考文献． 通过文献检索阅读, 揭示了以下有待探讨的理论问题： (a) 麦克斯韦-玻尔兹曼分布(Maxwell-Boltzmann distribution)的建立只涉及稀薄气体的压力内能,但未考虑粘性应力的内能;(b) 三个守恒律无法从玻尔兹曼方程直接导出,必须借助外加的小参数展开完成,同时在守恒方程中无法引入外力及能源的贡献;(c) Lattice Boltzmann Method (LBM)执行中,只更新流体的物质密度和平均速度,不更新其内能参数．由于在复杂流动中,流体的内能是时间及空间的函数,因此其理论是不完整的．以上揭示的理论问题是现有LBM方法不能有效地求解涉及高速及大压缩性引起内能剧烈变化的复杂流场的原因．作者给出一篇理论研究文章以回答揭示的理论问题．     

An Approach to Transport in Heterogeneous Porous Media Using the Truncated Temporal Moment Equations: Theory and Numerical Validation

In the last decade, the characterization of transport in porous media has benefited largely from numerical advances in applied mathematics and from the increasing power of computers. However, the resolution of a transport problem often remains cumbersome, mostly because of the time-dependence of the equations and the numerical stability constraints imposed by their discretization. To avoid these difficulties, another approach is proposed based on the calculation of the temporal moments of a curve of concentration versus time. The transformation into the Laplace domain of the transport equations makes it possible to develop partial derivative equations for the calculation of complete moments or truncated moments between two finite times, and for any point of a bounded domain. The temporal moment equations are stationary equations, independent of time, and with weaker constraints on their stability and diffusion errors compared to the classical advection–dispersion equation, even with simple discrete numerical schemes. Following the complete theoretical development of these equations, they are compared firstly with analytical solutions for simple cases of transport and secondly with a well-performing transport model for advective–dispersive transport in a heterogeneous medium with rate-limited mass transfer between the free water and an immobile phase. Temporal moment equations have a common parametrization with transport equations in terms of their parameters and their spatial distribution on a grid of discretization. Therefore, they can be used to replace the transport equations and thus accelerate the achievement of studies in which a large number of simulations must be carried out, such as the inverse problem conditioned with transport data or for forecasting pollution hazards.     

CADD: A seamless solution to the Domain Decomposition problem of subdomain boundaries and cross-points

The solution of wave problems using Domain Decomposition (DD) requires that the subdomain boundaries should be virtually non-existent, so that waves are not affected by the boundaries. This is a primary problem in DD, and it intensifies in the case of cross-points at which three or more subdomains meet. This topic has received a lot of attention in recent years, with special treatment of cross-points. This paper explains and demonstrates that this problem does not exist in Component-Averaged Domain Decomposition (CADD). CADD is implemented here with the authors’ CARP-CG algorithm, but it is shown that other implementations are also possible. The reason for the non-existence of this problem in CARP-CG is that in some superspace of the problem space, CARP-CG is mathematically equivalent to the CG acceleration of the Kaczmarz algorithm with cyclic relaxation parameters, applied to a single linear system. Due to its advantages, CARP-CG was adopted by some geophysics researchers as the solver of the Helmholtz and the elastic wave equations for full waveform inversion (FWI).