Showing papers on "Conservation law published in 1988"
TL;DR: The PSC algorithm as mentioned in this paper approximates the Hamilton-Jacobi equations with parabolic right-hand-sides by using techniques from the hyperbolic conservation laws, which can be used also for more general surface motion problems.
13,020 citations
01 Jan 1988
TL;DR: New numerical algorithms, called PSC algorithms, are devised for following fronts propagating with curvature-dependent speed, which approximate Hamilton-Jacobi equations with parabolic right-hand-sides by using techniques from the hyperbolic conservation laws.
Abstract: We devise new numerical algorithms, called PSC algorithms, for following fronts propagating with curvature-dependent speed. The speed may be an arbitrary function of curvature, and the front can also be passively advected by an underlying flow. These algorithms approximate the equations of motion, which resemble Hamilton-Jacobi equations with parabolic right-hand-sides, by using techniques from the hyperbolic conservation laws. Non-oscillatory schemes of various orders of accuracy are used to solve the equations, providing methods that accurately capture the formation of sharp gradients and cusps in the moving fronts. The algorithms handle topological merging and breaking naturally, work in any number of space dimensions, and do not require that the moving surface be written as a function. The methods can be also used for more general Hamilton-Jacobitype problems. We demonstrate our algorithms by computing the solution to a variety of surface motion problems.
2,501 citations
TL;DR: A deterministic collision rule for lattice-gas hydrodynamics that yields immiscible two-phase flow was proposed in this article. But the collision rule is based on a minimization principle and the conservation of mass, momentum, and particle type.
Abstract: We introduce a new deterministic collision rule for lattice-gas (cellular-automaton) hydrodynamics that yields immiscible two-phase flow. The rule is based on a minimization principle and the conservation of mass, momentum, and particle type. A numerical example demonstrates the spontaneous separation of two phases in two dimensions. Numerical studies show that the surface tension coefficient obeys Laplace's formula.
557 citations
TL;DR: In this article, a method of incorporating the space conservation law into a finite volume procedure is proposed and applied to a number of test cases, and the results show that the method is efficient and produces accurate results for all grid velocities and time steps for which temporal accuracy suffices.
Abstract: In the numerical solutions of fluid flow problems in moving co-ordinates, an additional conservation equation, namely the space conservation law, has to be solved simultaneously with the mass, momentum and energy conservation equations. In this paper a method of incorporating the space conservation law into a finite volume procedure is proposed and applied to a number of test cases. The results show that the method is efficient and produces accurate results for all grid velocities and time steps for which temporal accuracy suffices. It is also demonstrated, by analysis and test calculations, that not satisfying the space conservation law in a numerical solution procedure introduces errors in the form of artificial mass sources. These errors can be made negligible only by choosing a sufficiently small time step, which sometimes may be smaller than required by the temporal discretization accuracy.
353 citations
TL;DR: In this paper, it was shown that among a family of supersymmetric extensions of the Kortewegde Vries equation, there is a special system that has an infinite number of conservation laws, which can be formulated in the second Hamiltonian structure, and which has a nontrivial Lax representation.
Abstract: It is shown that among a one‐parameter family of supersymmetric extensions of the Korteweg–de Vries equation, there is a special system that has an infinite number of conservation laws, which can be formulated in the second Hamiltonian structure, and which has a nontrivial Lax representation. Its modified version is also discussed.
344 citations
TL;DR: In this paper, the boundary conditions for nonlinear hyperbolic systems of conservation laws were formulated based on the vanishing viscosity method and the Riemann problem, and the equivalence between these two conditions was studied.
265 citations
01 Jun 1988
TL;DR: In this paper, the boundary conditions for nonlinear hyperbolic systems of conservation laws were formulated based on the vanishing viscosity method and the Riemann problem, and the equivalence between these two conditions was studied.
Abstract: We propose two formulations of the boundary conditions for nonlinear hyperbolic systems of conservation laws. A first approach is based on the vanishing viscosity method and a second one is related to the Riemann problem. The equivalence between these two conditions is studied. The latter formulation is extended to treat numerically physically relevant boundary conditions. Monodimensional experiments are presented.
262 citations
01 Jan 1988
254 citations
TL;DR: In this article, a condition suffisante pour que a systeme se mette sous une forme normale d'un systeme hyperbolique-parabolique is defined.
Abstract: On considere le probleme aux valeurs initiales pour les systemes non lineaires d'ordre 2: w t +∑ j=1 n f j (w) x j =∑ i,j=1 n {G ij (w) w xj } x i ou G ij (w) sont des matrices m×m. On donne une condition suffisante pour que ce systeme se mette sous une forme normale d'un systeme hyperbolique-parabolique
167 citations
TL;DR: In this article, an unsplit, higher order Godunov method for scalar conservation laws in two dimensions is presented. But the method is not suitable for the special case of sharp fronts without any significant distortion.
159 citations
TL;DR: Convergence for total variation diminishing-second order resolution schemes approximating convex or concave conservation laws is shown by enforcing a single discrete entropy inequality.
Abstract: A unified treatment of explicit in time, two level, second order resolution, total variation diminishing, approximations to scalar conservation laws are presented. The schemes are assumed only to have conservation form and incremental form. A modified flux and a viscosity coefficient are introduced and results in terms of the latter are obtained. The existence of a cell entropy inequality is discussed and such an equality for all entropies is shown to imply that the scheme is an E scheme on monotone (actually more general) data, hence at most only first order accurate in general. Convergence for total variation diminishing-second order resolution schemes approximating convex or concave conservation laws is shown by enforcing a single discrete entropy inequality.
TL;DR: For nonlinear hyperbolic systems in non-conservation form, weak solutions in the class of bounded functions of bounded variation were considered in this article, and a generalized global entropy inequality was proposed and studied.
Abstract: For nonlinear hyperbolic systems in nonconservation form, we consider weak solutions in the class of bounded functions of bounded variation A generalized global entropy inequality is proposed and studied In this mathematical framework, we solve the Riemann problem and prove, for the Cauchy problem, the consistancy of the random choice method for systems in nonconservation form Our theory of entropy weak solutions is applied to nonconservative systems of elastodynamics and gasdynamics In particular, we give here a nonconservation form of the system of conservation laws of gasdynamics, which is equivalent for weak solutions in BV
TL;DR: The N = 2 superconformal algebra is related to the second hamiltonian structure of three integrable fermionic extensions of the Korteweg-de Vries equation as mentioned in this paper.
TL;DR: In this article, the bi-Hamiltonian structure for a large class of hyberbolic systems of conservation laws in two field variables, including the equations of gas dynamics, shallow water waves, one-dimensional elastic media, and the Born-Infeld equation from nonlinear electrodynamics, is exhibited.
Abstract: The bi‐Hamiltonian structure for a large class of one‐dimensional hyberbolic systems of conservation laws in two field variables, including the equations of gas dynamics, shallow water waves, one‐dimensional elastic media, and the Born–Infeld equation from nonlinear electrodynamics, is exhibited. For polytropic gas dynamics, these results lead to a quadri‐Hamiltonian structure. New higher‐order entropy‐flux pairs (conservation laws) and higher‐order symmetries are exhibited.
TL;DR: In this paper, a numerical method for first order nonlinear scalar hyperbolic conservation laws in one-dimensional space is presented, using an idea by Dafermos.
Abstract: A numerical method for first order nonlinear scalar hyperbolic conservation laws in one-dimension is presented, using an idea by Dafermos In this paper it is proved that it may be used as a numerical method for a general flux function and a general initial value It is possible to give explicit error estimates for the numerical method The error in the method is far smaller than in any other method The numerical method is illustrated in an example
TL;DR: In this paper, an uniqueness and existence theorem for the entropy weak solution of non-linear hyperbolic conservation laws with initial data and boundary condition was proved, which generalizes a result of Lax.
Abstract: We prove an uniqueness and existence theorem for the entropy weak solution of non-linear hyperbolic conservation laws of the form
, with initial data and boundary condition. The scalar function u = u(x, t), x > 0, t > 0, is the unknown; the function f = f(u) is assumed to be strictly convex. We also study the weighted Burgers' equation: α ϵ ℝ
.
We give an explicit formula, which generalizes a result of Lax. In particular, a free boundary problem for the flux f(u(.,.)) at the boundary is solved by introducing a variational inequality. The uniqueness result is obtained by extending a semigroup property due to Keyfitz.
TL;DR: Etude du comportement oscillatoire et de la convergence faible des solutions des approximations aux differences dispersives de u t +(1/2 u 2 ) x = 0 as mentioned in this paper.
Abstract: Etude du comportement oscillatoire et de la convergence faible des solutions des approximations aux differences dispersives de u t +(1/2 u 2 ) x =0
TL;DR: In this paper, the generalized Riemann problem for nonlinear hyperbolic systems of conservation laws is considered and the entropy solution of this problem is given in the form of an asymptotic expansion in time.
Abstract: We consider the generalized Riemann problem for nonlinear hyperbolic systems of conservation laws. We show in this paper that we can find the entropy solution of this problem in the form of an asymptotic expansion in time and we give an explicit method of construction of this asymptotic expansion. Finally, we define from this expansion an approximate solution of the generalized Riemann problem and we give error bounds.
TL;DR: The global Riemann problem for a nonstrictly hyperbolic system of conservation laws modeling polymer flooding is solved in this paper, where the system contains a term that models adsorption effects.
Abstract: The global Riemann problem for a nonstrictly hyperbolic system of conservation laws modeling polymer flooding is solved. In particular, the system contains a term that models adsorption effects.
31 Dec 1988
TL;DR: In this paper, the authors present a concise derivation of the time and volume-averaged conservation equations of two-phase flow in a form compatible with numerical evaluations using advanced generation, two fluid computer codes.
Abstract: The purpose of this paper is to present a concise derivation of the time and volume-averaged conservation equations of two-phase flow. These equations are in a form compatible with numerical evaluations using advanced generation, two fluid computer codes.
TL;DR: In this article, it was shown that the master equation describing fall-off effects in unimolecular and recombination reactions, with angular momentum conservation taken into account, can be solved exactly if the assumption is made that the probability of collisional energy transfer in J is independent of initial state.
Abstract: It is shown that the master equation describing fall-off effects in unimolecular and recombination reactions, with angular momentum (J) conservation taken into account, can be solved exactly if the assumption is made that the probability of collisional energy transfer in J is independent of initial state; this assumption is shown to be physically acceptable (from general conservation considerations and from trajectory calculations) for typical neutral radical recombination and decomposition reactions. This leads to a J-averaged master equation which can be readily solved by standard means. Illustrative computations using this treatment are presented.
TL;DR: In this paper, a method to derive conservation laws for evolution equations that describe pseudospherical surfaces is introduced based on a geometrical property of these surfaces and a new third-order evolution equation is obtained as a first example for a nongeneric case in the classification given by Chern and Tenenblat.
Abstract: A method to derive conservation laws for evolution equations that describe pseudospherical surfaces is introduced based on a geometrical property of these surfaces. A new third‐order evolution equation is obtained as a first example for a nongeneric case in the classification given by Chern and Tenenblat [Stud. Appl. Math. 74, 1 (1986)].
Journal Article•
TL;DR: On considere le probleme de Cauchy pour un systeme de lois de conservation strictement hyperbolique n×n:u t +(F(u)) x =0 u(x,0)=u 0 (x) as discussed by the authors, on demontre l'existence d'une solution faible definie pour tout t≥0.
Abstract: On considere le probleme de Cauchy pour un systeme de lois de conservation strictement hyperbolique n×n:u t +(F(u)) x =0 u(x,0)=u 0 (x). On demontre l'existence d'une solution faible definie pour tout t≥0. On etudie la possibilite d'utiliser des fonctionnelles continues de Glimm afin de construire une metrique, equivalente a la distance dans #7B-L 1 , qui est non dilatation par rapport au systeme
TL;DR: In this paper, a theory of second order diffusion processes and associated stochastic differential equations of second-order diffusion processes was developed. But the theory is not satisfied for a large class of reciprocal Gaussian processes.
Abstract: We develop a theory of second order diffusion processes and associated stochastic differential equations of second order. We show that equations of evolution of the density, mean velocity and momentum flux are a family of first order conservation laws similar to those of continuum mechanics. We verify that the theory is satisfied for a large class of reciprocal Gaussian processes
TL;DR: In this paper, the authors developed some remarkable new properties of the solutions of this system for resonant acoustics, including substantial almost periodic exchange of energy between the nonlinear sound waves, the existence of smooth periodic wave trains, and the role of such smooth wave patterns in eliminating or suppressing the strong temporal decay of sawtooth profile solutions of the decoupled inviscid Burgers equations.
Abstract: In general, weakly nonlinear high frequency almost periodic wave trains for systems of hyperbolic conservation laws interact and resonate to leading order In earlier work the first two authors and J Hunter developed simplified asymptotic equations describing this resonant interaction In the important special case of compressible fluid flow in one or several space dimensions, these simplified asymptotic equations are essentially two inviscid Burgers equations for the nonlinear sound waves, coupled by convolution with a known kernel given by the sum of the initial vortex strength and the derivative of the initial entropy Here we develop some of the remarkable new properties of the solutions of this system for resonant acoustics These new features include substantial almost periodic exchange of energy between the nonlinear sound waves, the existence of smooth periodic wave trains, and the role of such smooth wave patterns in eliminating or suppressing the strong temporal decay of sawtooth profile solutions of the decoupled inviscid Burgers equations Our approach combines detailed numerical modeling to elucidate the new phenomena together with rigorous analysis to obtain exact solutions as well as other elementary properties of the solutions of this system
TL;DR: In this paper, conservation laws for reflection and transmission of electromagnetic waves on a plane interface of isotropic transparent media are determined using these laws, relations have been established between the transverse shift (TS) of a centre of gravity of reflected and transmitted wavepackets, the change of the normal component of the intrinsic Minkowski angular momentum of the electromagnetic field and the Transverse electromagnetic power flow (TPF) phenomena.
Abstract: The conservation laws for the process of reflection and transmission of electromagnetic waves on a plane interface of isotropic transparent media are determined Using these laws, relations have been established between the transverse shift (TS) of a centre of gravity of reflected and transmitted wavepackets, the change of the normal component of the intrinsic Minkowski angular momentum of the electromagnetic field and the Abraham transverse momentum (or the transverse electromagnetic power flow (TPF)) The previous investigations of the TS and TPF phenomena are discussed from the point of view of conservation laws
01 Jan 1988
TL;DR: In this paper, a decomposition of the Bianchi identities in a Riemann-Cartan space-time with or without torsion is performed to determine those gravitational theories which have automatic conservation.
Abstract: Automatic conservation of energy-momentum and angular momentum is guaranteed in a gravitational theory if, via the field equations, the conservation laws for the material currents are reduced to the contracted Bianchi identities. We first execute an irreducible decomposition of the Bianchi identities in a Riemann-Cartan space-time. Then, starting from a Riemannian space-time with or without torsion, we determine those gravitational theories which have automatic conservation: general relativity and the Einstein-Cartan-Sciama-Kibble theory, both with cosmological constant, and the nonviable pseudoscalar model. The Poincare gauge theory of gravity, like gauge theories of internal groups, has no automatic conservation in the sense defined above. This does not lead to any difficulties in principle. Analogies to 3-dimensional continuum mechanics are stressed throughout the article.
TL;DR: In this paper, the joint droplet size and velocity distribution is derived by applying information theory to the atomization process, along with the normalization of the probability distribution function and the physical conservation laws of mass, momentum and energy.
Abstract: In this paper, the joint droplet size and velocity distribution is derived by applying information theory to the atomization process, along with the normalization of the probability distribution function and the physical conservation laws of mass, momentum and energy. The obtained distribution contains the Weber number as a variable, and agrees with experimental observations. An equation for the Sauter mean diameter (D32) is obtained which agrees with several of the expressions that have been obtained from correlations of experimental data. When the Weber number exceeds 4000, the results given by Li and Tankin (1987) are appropriate.
TL;DR: In this article, a Poincare-invariant scalar product and corresponding physical Hilbert space of states are constructed by finding a tensor current of rank 2, jμν(x1,x2), satisfying two independent conservation laws, relative to particles 1 and 2, respectively.
Abstract: In the framework of two‐particle relativistic quantum mechanics, a Poincare‐invariant scalar product and the corresponding physical Hilbert space of states are constructed. This is achieved by finding a tensor current of rank 2, jμν(x1,x2), satisfying two independent conservation laws, relative to particles 1 and 2, respectively. Then the scalar product is obtained by integrating the current jμν over two three‐dimensional spacelike hypersurfaces. The Hermiticity of the Poincare group generators is ensured by the fact that the kernel of the current jμν is translation invariant and covariant. A simple expression of the scalar product is obtained when one chooses for the two spacelike hypersurfaces two constant parallel hyperplanes. The positivity of the norm is, in general, ensured if the spectrum of the eigenvalues of the total mass squared operator comes out to be positive.
TL;DR: In this paper, a new fully second order scheme for convex scalar conservation laws is proposed, which is consistent with the one-sided Lipschitz condition (OSLC) that enforces both the entropy condition and their variation boundedness.
Abstract: Physical solutions to convex scalar conservation laws satisfy a one-sided Lipschitz condition (OSLC) that enforces both the entropy condition and their variation boundedness. Consistency with this condition is therefore desirable for a numerical scheme and was proved for both the Godunov and the Lax-Friedrichs scheme--also, in a weakened version, for the Roe scheme, all of them being only first order accurate. A new, fully second order scheme is introduced here, which is consistent with the OSLC. The modified equation is considered and shows interesting features. Another second order scheme is then considered and numerical results are discussed.