Showing papers on "Conservation law published in 1982"

Flux-vector splitting for the Euler equations

Abstract: When approximating a hyperbolic system of conservation laws w t + {f(w)} t = 0 with so-called upwind differences, we must, in the first place, establish which way the wind blows. More precisely, we must determine in which direction each of a variety of signals moves through the computational grid. For this purpose, a physical model of the interaction between computational cells is needed; at present two such models are in use.

Upwind difference schemes for hyperbolic systems of conservation laws

Abstract: We derive a new upwind finite difference approximation to systems of nonlinear hyperbolic conservation laws. The scheme has desirable properties for shock calculations. Under fairly general hypotheses we prove that limit solutions satisfy the entropy condition and that discrete steady shocks exist which are unique and sharp. Numerical examples involving the Euler and Lagrange equations of compressible gas dynamics in one and two space dimensions are given.

Formulas for determining local properties in molecular‐dynamics simulations: Shock waves

Abstract: Formulas are given that relate the mass, momentum, and energy densities and the momentum and heat fluxes to the masses, positions, and velocities of the individual particles making up a system. The formulas are similar to those of Irving and Kirkwood, but have forms that are easily implemented in molecular‐dynamics simulations. Even when simulating very inhomogeneous phenomena such as shock waves, the densities and fluxes exactly satisfy conservation laws for mass, momentum, and energy. Corrections to the virial formula for the pressure and to the related formulas for the stress tensor and heat flux are obtained. The relationship of the formulas given to those used by others is discussed.

Hamiltonian structure, symmetries and conservation laws for water waves

Abstract: An investigation on novel lines is made into the problem of water waves according to the perfect-fluid model, with reference to wave motions in both two and three space dimensions and with allowance for surface tension. Attention to the Hamiltonian structure of the complete nonlinear problem and the use of methods based on infinitesimal-transformation theory provide a Systematic account of symmetries inherent to the problem and of corresponding conservation laws.The introduction includes an outline of relevant elements from Hamiltonian theory (§ 1.1) and a brief discussion of implications that the present findings may carry for the approximate mathematical modelling of water waves (§1.2). Details of the hydrodynamic problem are recalled in §2. Then in §3 questions about the regularity of solutions are put in perspective, and a general interpretation is expounded regarding the phenomenon of wave-breaking as the termination of smooth Hamil- tonian evolution. In §4 complete symmetry groups are given for several versions of the water-wave problem : easily understood forms of the main results are listed first in §4.1, and the systematic derivations of them are explained in §4.2. Conservation laws implied by the one-parameter subgroups of the full symmetry groups are worked out in §5, where a recent extension of Noether's theorem is applied relying on the Hamiltonian structure of the problem. The physical meanings of the conservation laws revealed in §5, to an extent abstractly there, are examined fully in §6 and various new insights into the water-wave problem are presented.In Appendix 1 the parameterized version of the problem is considered, covering cases where the elevation of the free surface is not a single-valued function of horizontal position. I n Appendix 2 a general method for finding the symmetry groups of free-boundary problems is explained, and the exposition includes the mathematical material underlying the particular applications in §§4 and 5.

Glimm's Method for Gas Dynamics

Abstract: We investigate Glimm's method, a method for constructing approximate solutions to systems of hyperbolic conservation laws in one space variable by sampling explicit wave solutions It is extended to several space variables by operator splitting We consider two problems 1) We propose a highly accurate form of the sampling procedure, in one space variable, based on the van der Corput sampling sequence We test the improved sampling procedure numerically in the case of inviscid compressible flow in one space dimension and find that it gives high resolution results both in the smooth parts of the solution, as well as at discontinuities 2) We investigate the operator splitting procedure by means of which the multidimensional method is constructed An $O(1)$ error stemming from the use of this procedure near shocks oblique to the spatial grid is analyzed numerically in the case of the equations for inviscid compressible flow in two space dimensions We present a hybrid method which eliminates this error, consisting of Glimm's method, used in continuous parts of the flow, and the nonlinear Godunov method, used in regions where large pressure jumps are generated The resulting method is seen to be a substantial improvement over either of the component methods for multidimensional calculations

Kink dynamics in one-dimensional nonlinear systems

Abstract: A certain class of nonlinear evolution equations of one space dimension which permits kink type solutions and includes one-dimensional time-dependent Ginzburg-Landau (TDGL) equations and certain nonlinear wave equations is studied in some strong coupling approximation where the problem can be reduced to the study of kink dynamics. A detailed study is presented for the case of TDGL equation with possible applications to the late stage kinetics of order-disorer phase transitions and spinodal decompositions. A special case of kink dynamics of nonlinear wave equations is found to reduce to the Toda lattice dynamics. A new conservation law for dissipative systems is found which corresponds to the momentum conservation law for wave equations.

On the classical solutions of the initial value problem for the unmodified non-linear vlasov equation II special cases

Abstract: The theory of part I is applied to prove several existence and non-existence results for special cases. If N = 3, a global classical solution of P° exists, if φ ⩾ 0 has rotational symmetry. If N ⩾ 4, global classical solutions of P° do not always exist.

A note on uniqueness in thermoelasticity with one relaxation time

Abstract: Uniqueness for a general initial boundary-value problem of linear dynamic thermoelasticity with one relaxation time is established using the associated conservation law involving higher-order time derivatives.

Generalized Coordinate Forms of Governing Fluid Equations and Associated Geometrically Induced Errors

Abstract: The governing equations of fluid flow may be cast into various forms upon application of a generalized coordinate mapping. These forms are the nonconservation law form (NCLF) and strong, weak, and chain rule conservation law forms (SCLF, WCLF, CRCLF, respectively). This paper describes the geometrically induced errors resulting from the failure to satisfy a certain consistency condition for each of these four forms and also demonstrates the ability of the CRCLF to produce exactly the same numerical solution as the WCLF, provided a condition is met on where to evaluate the transformation metrics. It is also demonstrated that considerably fewer arithmetic operations are required to advance the solution from n to n +1 when the CRCLF is used in com- parison to both the SCLF and WCLF. HE effort devoted to the field of computational fluid dynamics is progressing at an ever-increasing rate. At certain points in time during this natural evolution of numerical methods, schemes, and ideas it is sometimes in- structive to pause to reflect upon past work in an attempt to gain the proper perspective. It is important that this reflection reaches all of the way to the fundamental rules and practices used to develop numerical techniques. Some of these practices are often taken for granted much too soon after their in- troduction into the literature. Only through a global view of this past effort can subtle, mutually experienced problem areas and possible causes be identified. The present work deals with the numerical solution to the transformed fluid flow equations (i.e., Euler equations, etc.) using the finite-difference approach. The transformed equations are obtained through application of a generalized mapping from physical coordinates to computational coordinates. At the time of application of the mapping to the original equations and prior to choosing the numerical in- tegration scheme, the analyst must decide in which form the equations should be written. The choices are the non- conservation law form (NCLF), strong conservation law form (SCLF), weak conservation law form (WCLF), and chain rule conservation law form (CRCLF). This decision is one of the fundamental practices alluded to in the previous paragraph. The present work illustrates how such a fundamental decision can strongly influence both the amount of analysis required to develop a consistent algorithm and the number of arithmetic operations required to execute the algorithm. In addition, it is shown that large solution errors and in some cases instabilities can result from failure to implement the results of such an analysis.

Transonic gas flow in a duct of varying area

Abstract: with w = (~, ~u, r c(x) : --a'(x)/a(x), etc. The purpose of this paper is to study the asymptotic states of general solutions of (1.1). In particular we consider the case when the flow is transonic and bifurcation occurs. The analysis is applicable to strictly hyperbolic systems of the more general form (1.1)'. By asymptotic states we mean solutions that produce no wave interactions. For a uniform duct, (I.1) ' reduces to the well known system of hyperbolic conservation laws

Waves on glaciers

Abstract: This paper is an attempt at a mathematical synopsis of the theory of wave motions on glaciers. These comprise surface waves (analogous to water waves) and seasonal waves (more like compression waves). Surface waves have been often treated and are well understood, but seasonal waves, while observed, do not seem to have attracted any theoretical explanation. Additionally, the spectacular phenomenon of glacier surges, while apparently a dynamic phenomenon, has not been satisfactorily explained. The present thesis is that the two wave motions (and probably also surging, though a discussion of this is not developed here) can both be derived from a rational theory based on conservation laws of mass and momentum, provided that the basal kinematic boundary condition involving boundary slip is taken to have a certain reasonable form. It is the opinion of this author that the form of this ‘sliding law’ is the crux of the difference between seasonal and surface waves, and that a further understanding of these motions must be based on a more satisfactory analysis of basal sliding. Since ice is here treated in the context of a slow, shallow, non-Newtonian fluid flow, the theory that emerges is that of non-Newtonian viscous shallow-water theory; rather than balance inertia terms with gravity in the momentum equation, we balance the shear-stress gradient. The resulting set of equationsis, in essence, a first-order nonlinear hyperbolic (kinematic) wave equation, and susceptible to various kinds of analysis. We show how both surface and seasonal waves are naturally described by such a model when the basal boundary condition is appropriately specified. Shocks can naturally occur, and we identify the (small) diffusive parameters that are present, and give the shock structure: in so doing, we gain a useful understanding of the effects of surface slope and longitudinal stress in these waves.

On the correspondence between symmetries and conservation laws of evolution equations

Abstract: There exists a well-defined correspondence between symmetries and conservation laws if an evolution equation admits a Hamiltonian formulation. We discuss whether the Hamiltonian structure is necessary for the correspondence.

The generalized Lagrangian-mean equations and hydrodynamic stability

Abstract: The generalized Lagrangian-mean (GLM) formulation of Andrews & McIntyre (1978 a , b ) offers alternative physical concepts and possible saving of effort in calculation, as compared with the more conventional Eulerian-mean approach. Though most existing applications of this theory concern waves on weakly sheared mean flows, it is also suitable for study of waves in strong shear flows. The hydrodynamic stability of parallel shear flows is examined from this point of view. An appreciation is gained of the roles of Stokes drift, pseudomomentum, energy and pseudoenergy in this context, such understanding being a necessary prerequisite for future developments. Several known results of linear stability theory, including the inflexion-point and semicircle theorems, are concisely rederived from the GLM conservation laws.

Momentum balance and flux conservation in model magnetospheric magnetic fields

Abstract: We have tested four recent quantitative magnetospheric magnetic field models to determine whether the magnetic field configuration is consistent with a balance of forces between the J x B force and the gradient of the pressure. A necessary condition for the models to be consistent with an isotropic pressure distribution is that curl (J x B) vanish. None of the models (Olson-Pfitzer, Tsyganenko, Hedgecock-Thomas, and Beard) meet this requirement: Our analysis indicates that the models are intrinsically inconsistent with momentum balance with any isotropic pressure distribution. We have also, for purposes of comparison, examined delxB. In certain models this can be nonzero.

Restricted Friedel sum rules and Korringa relations as consequences of conservation laws

Abstract: In a general model for impurities in a metallic host it is shown that separate Friedel sum rules hold for the total excess charge, spin and orbital momentum associated with the impurity. These relations are consequences of the conservation laws for charge, spin and orbital magnetic momentum, respectively. Furthermore, the Korringa relations are the exact consequences of the Ward identities, which are, in turn, consequences of some conservation laws in the Anderson model. Thus the Korringa relations are not valid for any of crystalline splitting, spin-orbit interaction and direct electron-electron interaction in the conduction band.