General Conservation Laws Research Articles

Essentially non-oscillatory Residual Distribution schemes for hyperbolic problems

The Residual Distribution (RD) schemes are an alternative to standard high order accurate finite volume schemes. They have several advantages: a better accuracy, a much more compact stencil, easy parallelization. However, they face several problems, at least for steady problems which are the only cases considered here. The solution is obtained via an iterative method. The iterative convergence must be good in order to get spatially accurate solutions, as suggested by the few theoretical results available for the RD schemes. In many cases, especially for systems, the iterative convergence is not sufficient to guaranty the theoretical accuracy. In fact, up to our knowledge, the iterative convergence is correct in only two cases: for first order monotone schemes and the (scalar) Struij’s PSI scheme which is a multidimensional upwind scheme. Up to our knowledge, the iterative convergence is poor for systems, except for the blended scheme of Deconinck et al. [Á. Csík, M. Ricchiuto, H. Deconinck, A conservative formulation of the multidimensional upwind residual distribution schemes for general nonlinear conservation laws, J. Comput. Phys. 179(2) (2002) 286–312] and Abgrall [R. Abgrall, Toward the ultimate conservative scheme: following the quest, J. Comput. Phys. 167(2) (2001) 277–315] which are also a genuinely multidimensional upwind scheme. A second drawback is that their construction relies, up to now, on a single first order scheme: the N scheme. However, it is known that standard first order finite volume schemes can be rephrased into a Residual Distribution framework. Unfortunately, the standard way of upgrading the order of accuracy to second order leads to very unsatisfactory results but clearly the construction of good schemes based on a wider class of first order schemes would be interesting. In this paper, we analyze these two problems, and show they are linked. We propose a fix and demonstrate its efficiency on several test cases that cover a wide range of applications. Our solution extends considerably the number of working RD schemes.

Error estimates for scalar conservation laws by a kinetic approach

We use the kinetic approach of Perthame and Tadmor (1991) to calculate the error estimates for general scalar conservation laws governing problems in gas dynamics or fluid mechanics in general. The Kružkov and Kuznetsov techniques are generalized to this method, and an error bound of order (where ε is the mean free path) is obtained.

Spectral (finite) volume method for conservation laws on unstructured grids V: Extension to three-dimensional systems

In this paper, the fifth in a series, the high-order spectral finite-volume, or spectral volume (SV) method for unstructured grids is extended to three dimensions. Limitations of conventional structured and unstructured methods are first reviewed. The spectral finite-volume method for generalized conservation laws is then described. It is shown that if all grid cells are partitioned into structured sub-cells in a similar manner, the discretizations become universal, and are reduced to the same weighted sum of unknowns involving just a few simple adds and multiplies. Important aspects of the data structure and its effects on communication and the optimum use of cache memory are discussed. Previously defined one-parameter partitions of the SV in 2D are extended to multiple parameters and then used to construct 3D partitions. Numerical solutions of plane electromagnetic waves incident on perfectly conducting circular cylinders and spheres are presented and compared with the exact solution to demonstrate the capability of the method. Excellent agreement has been found. Computation timings show that the new method is more efficient than conventional structured and unstructured methods.

Residual distribution for general time-dependent conservation laws

We consider the second-order accurate numerical solution of general time-dependent hyperbolic conservation laws over unstructured grids in the framework of the Residual Distribution method. In order to achieve full conservation of the linear, monotone and first-order space-time schemes of (Csik et al., 2003) and (Abgrall et al., 2000), we extend the conservative residual distribution (CRD) formulation of (Csik et al., 2002) to prismatic space-time elements. We then study the design of second-order accurate and monotone schemes via the nonlinear mapping of the local residuals of linear monotone schemes. We derive sufficient and necessary conditions for the well-posedness of the mapping. We prove that the schemes obtained with the CRD formulation satisfy these conditions by construction. Thus the nonlinear schemes proposed in this paper are always well defined. The performance of the linear and nonlinear schemes are evaluated on a series of test problems involving the solution of the Euler equations and of a two-phase flow model. We consider the resolution of strong shocks and complex interacting flow structures. The results demonstrate the robustness, accuracy and non-oscillatory character of the proposed schemes. d schemes.

Convergence des méthodes particulaires renormalisées pour les systèmes de Friedrichs

We present a study of the renormalized particle scheme. Renormalization is a tool introduced in order to alleviate the SPH particle methods' lack of consistency. A conservative scheme, the weak renormalized scheme, is derived from the general conservation laws weak formulation. We apply this scheme to Friedrichs systems. The weak renormalized scheme being unstable, we introduce a numerical viscosity before applying an explicit Euler time discretization, and thus construct the numerical scheme whose convergence in L 2 norm is studied. To cite this article: N. Lanson, J.-P. Vila, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

Electrodynamic analogue of detonation waves in a ferroelectric transmission line

The formation of electrodynamic shock waves in non-linear transmission lines is an experimentally demonstrated phenomenon. Theoretical treatment of such phenomena has so far been limited to device-specific models. In this paper, a general hyperbolic conservation law formalism has been developed to describe non-linear propagation of electrical pulses in arbitrary transmission lines. This enables the use of powerful mathematical methods developed in the context of hyperbolic systems of partial differential equations.This formalism is used to examine the possibility of an electrodynamic analogue of detonation wave in a ferroelectric transmission line. A poled ferroelectric material forming the dielectric of a transmission line not only acts as a non-linear medium with a voltage-dependent dielectric constant but also as a charge and energy storage element. An electric pulse greater than the coercive field injected into this transmission line would steepen during its propagation because of the non-linear, voltage-dependent propagation velocity and also depolarize the ferroelectric material as it travels, adding charge and energy to the wave and sustaining it in the face of losses in a manner reminiscent of detonation waves in a reactive energetic material.

The space of general conservation laws and topology changes

We construct a 5D space in which conservation laws are always and absolutely valid. This general result holds also scaling in 4D space even inducing, when necessary, changes of topology. The inequality of representations of physical laws in 4D and in 5D generalized spaces is founded not on phenomenological but on mathematical results. Essentially, the mechanism is based on the fact that we take into account also transformations where the Jacobian of the change of variables can assume null values. We derive that stress-energy and metric tensors, due to Bianchi identities, are always conserved through the mechanism presented in the following based on the fact that we introduce transformations with null Jacobian in the space definition. Moreover, in the case of metric tensor, conservation is a necessary feature in order to have physically defined theories.

Implicit space–time residual distribution method for unsteady laminar viscous flow

The class of multidimensional upwind residual distribution (RD) schemes has been developed in the past decades as an attractive alternative to the finite volume (FV) and finite element (FE) approaches. Although they have shown superior performances in the simulation of steady two-dimensional and three-dimensional inviscid and viscous flows, their extension to the simulation of unsteady flow fields is still a topic of intense research [ICCFD2, International Conference on Computational Fluid Dynamics 2, Sydney, Australia, 15–19 July 2002; M. Mezine, R. Abgrall, Upwind multidimensional residual schemes for steady and unsteady flows]. Recently the space–time RD approach has been developed by several researchers [Int. J. Numer. Methods Fluids 40 (2002) 573; J. Comput. Phys. 188 (2003) 16; Á.G. Csı́k, Upwind residual distribution schemes for general hyperbolic conservation laws and application to ideal magnetohydrodynamics, PhD thesis, Katholieke Universiteit Leuven, 2002; J. Comput. Phys. 188 (2003) 16; R. Abgrall; M. Mezine, Construction of second order accurate monotone and stable residual distribution schemes for unsteady flow problems] which allows to perform second order accurate unsteady inviscid computations. In this paper we follow the work done in [Int. J. Numer. Methods Fluids 40 (2002) 573; Á.G. Csı́k, Upwind residual distribution schemes for general hyperbolic conservation laws and application to ideal magnetohydrodynamics, PhD thesis, Katholieke Universiteit Leuven, 2002]. In this approach the space–time domain is discretized and solved as a ( d+1)-dimensional problem, where d is the number of space dimensions. In [Int. J. Numer. Methods Fluids 40 (2002) 573; Á.G. Csı́k, Upwind residual distribution schemes for general hyperbolic conservation laws and application to ideal magnetohydrodynamics, PhD thesis, Katholieke Universiteit Leuven, 2002] it is shown that thanks to the multidimensional upwinding of the RD method, the solution of the unsteady problem can be decoupled into sub-problems on space–time slabs composed of simplicial elements, allowing to obtain a true time marching procedure. Moreover, the method is implicit and unconditionally stable for arbitrary large time-steps if positive RD schemes are employed. We present further development of the space–time approach of [Int. J. Numer. Methods Fluids 40 (2002) 573; Á.G. Csı́k, Upwind residual distribution schemes for general hyperbolic conservation laws and application to ideal magnetohydrodynamics, PhD thesis, Katholieke Universiteit Leuven, 2002] by extending it to laminar viscous flow computations. A Petrov–Galerkin treatment of the viscous terms [Project Report 2002-06, von Karman Institute for Fluid Dynamics, Belgium, 2002; J. Dobeš, Implicit space–time method for laminar viscous flow], consistent with the space–time formulation has been investigated, implemented and tested. Second order accuracy in both space and time was observed on unstructured triangulation of the spatial domain. The solution is obtained at each time-step by solving an implicit non-linear system of equations. Here, following [Int. J. Numer. Methods Fluids 40 (2002) 573; Á.G. Csı́k, Upwind residual distribution schemes for general hyperbolic conservation laws and application to ideal magnetohydrodynamics, PhD thesis, Katholieke Universiteit Leuven, 2002], we formulate the solution of this system as a steady state problem in a pseudo-time variable. We discuss the efficiency of an explicit Euler forward pseudo-time integrator compared to the implicit Euler. When applied to viscous computation, the implicit method has shown speed-ups of more than a factor 50 in terms of computational time.

Variational description of multifluid hydrodynamics: Uncharged fluids

We present a formalism for Newtonian multifluid hydrodynamics derived from an unconstrained variational principle. This approach provides a natural way of obtaining the general equations of motion for a wide range of hydrodynamic systems containing an arbitrary number of interacting fluids and superfluids. In addition to spatial variations we use ``time shifts'' in the variational principle, which allows us to describe dissipative processes with entropy creation, such as chemical reactions, friction or the effects of external non-conservative forces. The resulting framework incorporates the generalization of the entrainment effect originally discussed in the case of the mixture of two superfluids by Andreev and Bashkin. In addition to the conservation of energy and momentum, we derive the generalized conservation laws of vorticity and helicity, and the special case of Ertel's theorem for the single perfect fluid. We explicitly discuss the application of this framework to thermally conducting fluids, superfluids, and superfluid neutron star matter. The equations governing thermally conducting fluids are found to be more general than the standard description, as the effect of entrainment usually seems to be overlooked in this context. In the case of superfluid ${}^{4}\mathrm{He}$ we recover the Landau-Khalatnikov equations of the two-fluid model via a translation to the ``orthodox'' framework of superfluidity, which is based on a rather awkward choice of variables. Our two-fluid model for superfluid neutron star matter allows for dissipation via mutual friction and also ``transfusion'' via $\ensuremath{\beta}$ reactions between the neutron fluid and the proton-electron fluid.

Error Estimates to Smooth Solutions of Runge--Kutta Discontinuous Galerkin Methods for Scalar Conservation Laws

In this paper we study the error estimates to sufficiently smooth solutions of scalar conservation laws for Runge--Kutta discontinuous Galerkin (RKDG) methods, where the time discretization is the second order explicit total variation diminishing (TVD) Runge--Kutta method. Error estimates for the $\mathbb{P}^1$ (piecewise linear) elements are obtained under the usual CFL condition $\tau\leq \gamma h$ for general nonlinear conservation laws in one dimension and for linear conservation laws in multiple space dimensions, where h and $\tau$ are the maximum element lengths and time steps, respectively, and the positive constant $\gamma$ is independent of h and $\tau$. However, error estimates forhigher order $\mathbb{P}^k(k\geq 2)$ elements need a more restrictive time step $\tau\leq \gamma h^{4/3}$. We remark that this stronger condition is indeed necessary, as the method is linearly unstable under the usual CFL condition $\tau\leq\gamma h$ for the $\mathbb{P}^k$ elements of degree $k\geq 2$. Error estimates of $O(h^{k+1/2}+\tau^2)$ are obtained for general monotone numerical fluxes, and optimal error estimates of $O(h^{k+1}+\tau^2)$ are obtained for upwind numerical fluxes.

A Kinetic Formulation for Multidimensional Scalar Conservation Laws with Boundary Conditions and Applications

We state a kinetic formulation of weak entropy solutions of a general multidimensional scalar conservation law with initial and boundary conditions. We first associate with any weak entropy solution an entropy defect measure; the analysis of this measure at the boundary of the domain relies on the study of weak entropy sub- and supersolutions and implies the introduction of the notion of sided boundary defect measures. As a first application, we prove that any weak entropy subsolution of the initial-boundary value problem is bounded above by any weak entropy supersolution (comparison theorem). We next study a Bhatnagar--Gross--Krook-like kinetic model that approximates the scalar conservation law. We prove that such a model converges by adapting the proof of the comparison theorem.

A New Approach to Noncommutative Soliton Equations

A geometric description is introduced to consistently define initial and/or boundary value problem, zero curvature representation and monodromy operator for noncommutative (nc, for short) soliton equations. Based on the description, it is shown that many known commutative/noncommutative soliton equations can be obtained in a unified way. The integrability of a nc-NLS equation and a generalized large N-limit of matrix NLS equation is studied; a generalized conservation law for the nc-NLS equation is presented, and the generalized large N-limit of matrix NLS equation is shown to be integrable in the sense that there exist an infinite number of independent conserved quantities in involution.

A Dynamical Unification Scheme from General Conservation Laws

The aim of this work is to present an unification scheme of fundamental interactions based on a well defined dynamics, the non-introduction of ad hoc hypotheses and the consideration of the minimal necessary number of free parameters and dimensions. A dynamical unification scheme of fundamental interactions can be achieved assuming a 5D space where conservation laws are always and absolutely valid, i.e. never violated. This approach gives rise to an induced-matter theory in the usual 4D space-time through a process of embedding and dimensional reduction by which masses, spins and charges of particles naturally spring out, and also the the hierarchy problem can be successfully faced thanks to the mass spectrum. The emergence of asymptotic freedom also for gravitational interaction, the existence of two time arrows together with the possibility of closed time-like paths are intrinsic results of such a theory, leading to a recovering of the causality principle and to a formal, dynamical explanation of several paradoxes and questioning problems of modern physics e.g. entanglement of EPR-type quantum states, quantum teleportation, gamma ray bursts origin, black hole singularities and cosmic primary antimatter absence.

Scaling Approach to the Growth Equation with a GeneralizedConservation Law

The Flory-type scaling approach proposed by Hentscheland Family [Phys. Rev. Lett. 66 (1991) 1982] is generalized tothe analysis of the growth equation with a generalized conservationlaw, which contains the Kardar–Parisi–Zhang, Sun–Guo–Grant, andmolecular-beam epitaxy growth equations as special cases and allows fora unified investigation of growth equations. The scaling exponentsobtained here can be in agreement well with the corresponding resultsderived by the dynamic renormalization group theory and the previousscaling analyses.

Gamma ray bursts as a signature for entangled gravitational systems

Gamma ray bursts (GRBs), due to their features, can be considered not only extremely energetic, but also as the most relativistic astrophysical objects discovered. Their phenomenology is still matter of debate and, till now, no fully satisfactory model has been formulated to explain the nature of their origin. In the framework of a recently developed new theory, where general conservation laws are always and absolutely conserved in nature, we propose an alternative model where an “entangled” gravitational system, dynamically constituted by a black holes connected to a white hole through a worm hole, seems capable of explaining most of the properties inferred for the GRB engine. In particular, it leads to a natural explanation of energetics, beaming, polarization, and, very likely, distribution. On the other hand, GRBs can be considered a signature of such entangled gravitational systems.

A Conservative Formulation of the Multidimensional Upwind Residual Distribution Schemes for General Nonlinear Conservation Laws

In the present paper we consider the numerical solution of systems of general nonlinear hyperbolic conservation laws on unstructured grids by means of the residual distribution method. We propose a new formulation of the first-order linear, optimal positive N scheme, relying on a contour integration of the convective fluxes over the boundaries of an element. Full conservation is achieved for arbitrary flux functions, while the robustness and the monotone shock capturing of the original N scheme is retained. The new variant of the N scheme is combined with the conservative second-order linear LDA scheme to obtain a nonlinear second-order monotone B scheme. The performance of the new residual distribution schemes is evaluated on problems governed by the Euler equations. As an application to a more complex system of conservation laws lacking an exact conservative linearization, we solve the ideal magnetohydrodynamics equations in two spatial dimensions.

Convergence rates to viscous shock profile for general scalar viscous conservation laws with large initial disturbance

This paper is concerned with the convergence rates to viscous shock profile for general scalar viscous conservation laws. Compared with former results in this direction, the main novelty in this paper lies in the fact that the initial disturbance can be chosen arbitrarily large, This answers positively an open problem proposed by A. Matsumura in [12] and K. Nishihara in [16]. Our analysis is based on the L-1-stability results obtained by H. Freistuhler and D. Serre in [1].

Restless bandits, partial conservation laws and indexability

We show that if performance measures in a general stochastic scheduling problem satisfy partial conservation laws (PCL), which extend the generalized conservation laws (GCL) introduced by Bertsimas and Nino-Mora (1996), then the problem is solved optimally by a priority-index policy under a range of admissible linear performance objectives, with both this range and the optimal indices being determined by a one-pass adaptive-greedy algorithm that extends Klimov’s: we call such scheduling problems PCL-indexable. We further apply the PCL framework to investigate the indexability property of restless bandits (two-action finite-state Markov decision chains) introduced by Whittle, obtaining the following results: (i) we present conditions on model parameters under which a single restless bandit is PCL-indexable, and hence indexable; membership ofthe class of PCL-indexable bandits is tested through a single run ofthe adaptive-greedy algorithm, which further computes the Whittle indices when the test is positive; this provides a tractable sufficient condition for indexability; (ii) we further introduce the subclass of GCL-indexable bandits (including classical bandits), which are indexable under arbitrary linear rewards. Our analysis is based on the achievable region approach to stochastic optimization, as the results follow from deriving and exploiting a new linear programming reformulation for single restless bandits.

Restless bandits, partial conservation laws and indexability

We show that if performance measures in a general stochastic scheduling problem satisfy partial conservation laws (PCL), which extend the generalized conservation laws (GCL) introduced by Bertsimas and Niño-Mora (1996), then the problem is solved optimally by a priority-index policy under a range ofadmissiblelinear performance objectives, with both this range and the optimal indices being determined by a one-pass adaptive-greedy algorithm that extends Klimov's: we call such scheduling problemsPCL-indexable. We further apply the PCL framework to investigate the indexability property of restless bandits (two-action finite-state Markov decision chains) introduced by Whittle, obtaining the following results: (i) we present conditions on model parameters under which a single restless bandit is PCL-indexable, and hence indexable; membership of the class of PCL-indexable bandits is tested through a single run of the adaptive-greedy algorithm, which further computes the Whittle indices when the test is positive; this provides a tractable sufficient condition for indexability; (ii) we further introduce the subclass ofGCL-indexablebandits (including classical bandits), which are indexable under arbitrary linear rewards. Our analysis is based on the achievable region approach to stochastic optimization, as the results follow from deriving and exploiting a new linear programming reformulation for single restless bandits.

Methods for the Elimination of Unknowns from Systems of Linear Inequalities and Their Applications

We study methods for the elimination of an unknown or a group of unknowns from systems of linear inequalities. We justify these methods by using the Helly theorem. The methods considered are applied to the calculation of streams in networks with a generalized conservation law.