Showing papers on "Conservation law published in 2012"

Arbitrarily High-order Accurate Entropy Stable Essentially Nonoscillatory Schemes for Systems of Conservation Laws

Abstract: We design arbitrarily high-order accurate entropy stable schemes for systems of conservation laws. The schemes, termed TeCNO schemes, are based on two main ingredients: (i) high-order accurate entropy conservative fluxes and (ii) suitable numerical diffusion operators involving ENO reconstructed cell-interface values of scaled entropy variables. Numerical experiments in one and two space dimensions are presented to illustrate the robust numerical performance of the TeCNO schemes.

Maximum-Principle-Satisfying and Positivity-Preserving High Order Discontinuous Galerkin Schemes for Conservation Laws on Triangular Meshes

Abstract: In Zhang and Shu (J. Comput. Phys. 229:3091---3120, 2010), two of the authors constructed uniformly high order accurate finite volume and discontinuous Galerkin (DG) schemes satisfying a strict maximum principle for scalar conservation laws on rectangular meshes. The technique is generalized to positivity preserving (of density and pressure) high order DG or finite volume schemes for compressible Euler equations in Zhang and Shu (J. Comput. Phys. 229:8918---8934, 2010). The extension of these schemes to triangular meshes is conceptually plausible but highly nontrivial. In this paper, we first introduce a special quadrature rule which is exact for two-variable polynomials over a triangle of a given degree and satisfy a few other conditions, by which we can construct high order maximum principle satisfying finite volume schemes (e.g. essentially non-oscillatory (ENO) or weighted ENO (WENO) schemes) or DG method solving two dimensional scalar conservation laws on triangular meshes. The same method can preserve the maximum principle for DG or finite volume schemes solving two-dimensional incompressible Euler equations in the vorticity stream-function formulation, or any passive convection equation with an incompressible velocity field. We also obtain positivity preserving (for density and pressure) high order DG or finite volume schemes solving compressible Euler equations on triangular meshes. Numerical tests for the third order Runge-Kutta DG (RKDG) method on unstructured meshes are reported.

Hamiltonian dynamics and Noether symmetries in Extended Gravity Cosmology

Abstract: We discuss the Hamiltonian dynamics for cosmologies coming from Extended Theories of Gravity. In particular, minisuperspace models are taken into account searching for Noether symmetries. The existence of conserved quantities gives selection rule to recover classical behavior in cosmic evolution according to the so-called Hartle criterion, which allows one to select correlated regions in the configuration space of dynamical variables. We show that such a statement works for general classes of Extended Theories of Gravity and is conformally preserved. Furthermore, the presence of Noether symmetries allows a straightforward classification of singularities that represent the points where the symmetry is broken. Examples for non-minimally coupled and higher-order models are discussed.

Well-Posedness in Smooth Function Spaces for the Moving-Boundary Three-Dimensional Compressible Euler Equations in Physical Vacuum

Abstract: We prove well-posedness for the three-dimensional compressible Euler equations with moving physical vacuum boundary, with an equation of state given by p(ρ) = Cγργ for γ > 1. The physical vacuum singularity requires the sound speed c to go to zero as the square-root of the distance to the moving boundary, and thus creates a degenerate and characteristic hyperbolic free-boundary system wherein the density vanishes on the free-boundary, the uniform Kreiss–Lopatinskii condition is violated, and manifest derivative loss ensues. Nevertheless, we are able to establish the existence of unique solutions to this system on a short time-interval, which are smooth (in Sobolev spaces) all the way to the moving boundary, and our estimates have no derivative loss with respect to initial data. Our proof is founded on an approximation of the Euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term, chosen to preserve as much of the geometric structure of the Euler equations as possible. We first construct solutions to this degenerate parabolic regularization using a higher-order version of Hardy’s inequality; we then establish estimates for solutions to this degenerate parabolic system which are independent of the artificial viscosity parameter. Solutions to the compressible Euler equations are found in the limit as the artificial viscosity tends to zero. Our regular solutions can be viewed as degenerate viscosity solutions. Our methodology can be applied to many other systems of degenerate and characteristic hyperbolic systems of conservation laws.

Concentration Compactness for Critical Wave Maps

Abstract: Wave maps are the simplest wave equations taking their values in a Riemannian manifold (M,g). Their Lagrangian is the same as for the scalar equation, the only difference being that lengths are measured with respect to the metric g. By Noether's theorem, symmetries of the Lagrangian imply conservation laws for wave maps, such as conservation of energy. In coordinates, wave maps are given by a system of semilinear wave equations. Over the past 20 years important methods have emerged which address the problem of local and global wellposedness of this system. Due to weak dispersive effects, wave maps defined on Minkowski spaces of low dimensions, such as $R_{t,x}^{2+1}$, present particular technical difficulties. This class of wave maps has the additional important feature of being energy critical, which refers to the fact that the energy scales exactly like the equation. Around 2000 Daniel Tataru and Terence Tao, building on earlier work of Klainerman–Machedon, proved that smooth data of small energy lead to global smooth solutions for wave maps from 2+1 dimensions into target manifolds satisfying some natural conditions. In contrast, for large data, singularities may occur in finite time for $M=S^2$ as target. This monograph establishes that for $H$ as target the wave map evolution of any smooth data exists globally as a smooth function. While we restrict ourselves to the hyperbolic plane as target the implementation of the concentration-compactness method, the most challenging piece of this exposition, yields more detailed information on the solution. This monograph will be of interest to experts in nonlinear dispersive equations, in particular to those working on geometric evolution equations.

Sparse tensor multi-level Monte Carlo finite volume methods for hyperbolic conservation laws with random initial data

Abstract: We consider scalar hyperbolic conservation laws in several (d ≥ 1) spatial dimensions with stochastic initial data. We prove existence and uniqueness of a random-entropy solution and show existence of statistical moments of any order k of this random entropy solution. We present a class of numerical schemes of multi-level Monte Carlo Finite Volume (MLMC-FVM) type for the approximation of random entropy solutions as well as of their k-point correlation functions. These schemes are shown to obey the same accuracy vs. work estimate as a single application of the finite volume solver for the corresponding deterministic problem. Numerical experiments demonstrating the efficiency of these schemes are presented. Statistical moments of discontinuous solutions are found to be more regular than pathwise solutions.

SPHS: smoothed particle hydrodynamics with a higher order dissipation switch

Abstract: We present a novel implementation of smoothed particle hydrodynamics that uses the spatial derivative of the velocity divergence as a higher order dissipation switch. Our switch – which is second order accurate – detects flow convergence before it occurs. If particle trajectories are going to cross, we switch on the usual SPH artificial viscosity, as well as conservative dissipation in all advected fluid quantities (e.g. the entropy). The viscosity and dissipation terms (that are numerical errors) are designed to ensure that all fluid quantities remain single valued as particles approach one another, to respect conservation laws, and to vanish on a given physical scale as the resolution is increased. SPHS alleviates a number of known problems with ‘classic’ SPH, successfully resolving mixing, and recovering numerical convergence with increasing resolution. An additional key advantage is that – treating the particle mass similarly to the entropy – we are able to use multimass particles, giving significantly improved control over the refinement strategy. We present a wide range of code tests including the Sod shock tube, Sedov–Taylor blast wave, Kelvin–Helmholtz Instability, the ‘blob test’ and some convergence tests. Our method performs well on all tests, giving good agreement with analytic expectations.

A class of nonlocal models for pedestrian traffic

Abstract: We present a new class of macroscopic models for pedestrian flows. Each individual is assumed to move towards a fixed target, deviating from the best path according to the instantaneous crowd distribution. The resulting equation is a conservation law with a nonlocal flux. Each equation in this class generates a Lipschitz semigroup of solutions and is stable with respect to the functions and parameters defining it. Moreover, key qualitative properties such as the boundedness of the crowd density are proved. Specific models are presented and their qualitative properties are shown through numerical integrations. In particular, the present model accounts for the possibility of reducing the exit time from a room by carefully positioning obstacles that direct the crowd flow.

Mechanics in Material Space: with Applications to Defect and Fracture Mechanics

Abstract: 1 Mathematical Preliminaries.- 1.1 General Remarks.- 1.2 What is a Conservation Law?.- 1.3 Trivial Conservation Laws.- 1.4 System with a Lagrangian Noether's Method.- 1.5 System without a Lagrangian Neutral-Action Method.- 1.6 Discussion.- 2 Linear Theory of Elasticity.- 2.1 General Remarks.- 2.2 Elements of Linear Elasticity.- 2.3 Conservation Laws of Linear Elastostatics.- 2.4 Alternative Derivations of Conservation Laws.- 3 Properties of the Eshelby Tensor.- 3.1 General Remarks 81.- 3.2 Physical Interpretation of the Components of the Eshelby Tensor.- 3.3 Invariants, Principal Values, Principal Directions and Extremal Values of the Eshelby Tensor.- 4 Linear Elasticity with Defects.- 4.1 General Remarks.- 4.2 Path-Independent Integrals and Energy-Release Rates.- 4.3 Example: Hole-Dislocation Interaction.- 4.4 Path-Independent Integrals of Fracture Mechanics.- 5 Inhomogeneous Elastostatics.- 5.1 General Remarks.- 5.2 Symmetry Transformations.- 5.3 The Homogeneous Case.- 5.4 The Inhomogeneous Case.- 5.5 Relation to Stress-Intensity Factors.- 5.6 Examples.- 6 Elastodynamics.- 6.1 General Remarks.- 6.2 Time t as an Additional Independent Variable.- 6.3 Convolution in Time.- 6.4 Domain-Independent Integrals.- 6.5 Energy-Release Rates.- 6.6 Wave Motion.- 7 Dissipative Systems.- 7.1 General Remarks.- 7.2 Diffusion Equation.- 7.3 Non-Linear Wave Equation.- 7.4 Viscoelasticity.- 8 Coupled Fields.- 8.1 General Remarks.- 8.2 Piezoelectricity.- 8.3 Thermoelasticity.- 8.4 Mechanics of a Porous Medium.- 9 Bars, Shafts and Beams.- 9.1 General Remarks.- 9.2 Elements of Strength-of-Materials.- 9.3 Balance and Conservation Laws for Bars and Shafts.- 9.4 Balance and Conservation Laws for Beams.- 9.5 Energy-Release Rates and Stress-Intensity Factors.- 9.6 Examples.- 10 Plates and Shells.- 10.1 General Remarks.- 10.2 Plate Theories.- 10.3 Conservation Laws for Elastostatics of Mindlin Plates.- 10.4 Reduction to the Classical Theory.- 10.5 Conservation Laws for Shells.- Appendix A.- Conservation Laws for Inhomogeneous Bars under Arbitrary Axial Loading.- Appendix B.- B.1 Elastodynamics of Inhomogeneous Bernoulli-Euler Beams.- B.2 Reduction to Statics.- Appendix C.- C.1 Elastodynamics of Inhomogeneous Mindlin Plates.- C.2 Reduction to Statics.- References.- Symbol Index.- Author Index.

Numerical simulations of Hall-effect plasma accelerators on a magnetic-field-aligned mesh.

Abstract: The ionized gas in Hall-effect plasma accelerators spans a wide range of spatial and temporal scales, and exhibits diverse physics some of which remain elusive even after decades of research. Inside the acceleration channel a quasiradial applied magnetic field impedes the current of electrons perpendicular to it in favor of a significant component in the E×B direction. Ions are unmagnetized and, arguably, of wide collisional mean free paths. Collisions between the atomic species are rare. This paper reports on a computational approach that solves numerically the 2D axisymmetric vector form of Ohm's law with no assumptions regarding the resistance to classical electron transport in the parallel relative to the perpendicular direction. The numerical challenges related to the large disparity of the transport coefficients in the two directions are met by solving the equations on a computational mesh that is aligned with the applied magnetic field. This approach allows for a large physical domain that extends more than five times the thruster channel length in the axial direction and encompasses the cathode boundary where the lines of force can become nonisothermal. It also allows for the self-consistent solution of the plasma conservation laws near the anode boundary, and for simulations in accelerators with complex magnetic field topologies. Ions are treated as an isothermal, cold (relative to the electrons) fluid, accounting for the ion drag in the momentum equation due to ion-neutral (charge-exchange) and ion-ion collisions. The density of the atomic species is determined using an algorithm that eliminates the statistical noise associated with discrete-particle methods. Numerical simulations are presented that illustrate the impact of the above-mentioned features on our understanding of the plasma in these accelerators.

From discrete particles to continuum fields near a boundary

Abstract: An expression for the stress tensor near an external boundary of a discrete mechanical system is derived explicitly in terms of the constituents’ degrees of freedom and interaction forces. Starting point is the exact and general coarse graining formulation presented by Goldhirsch (Granul Mat 12(3):239–252, 2010), which is consistent with the continuum equations everywhere but does not account for boundaries. Our extension accounts for the boundary interaction forces in a self-consistent way and thus allows the construction of continuous stress fields that obey the macroscopic conservation laws even within one coarse-graining width of the boundary. The resolution and shape of the coarse-graining function used in the formulation can be chosen freely, such that both microscopic and macroscopic effects can be studied. The method does not require temporal averaging and thus can be used to investigate time-dependent flows as well as static or steady situations. Finally, the fore-mentioned continuous field can be used to define ‘fuzzy’ (very rough) boundaries. Discrete particle simulations are presented in which the novel boundary treatment is exemplified, including chute flow over a base with roughness greater than one particle diameter.

Compact reconstruction schemes with weighted eno limiting for hyperbolic conservation laws

Abstract: The simulation of turbulent compressible flows requires an algorithm with high ac- curacy and spectral resolution to capture different length scales, as well as nonoscillatory behavior across discontinuities like shock waves. Compact schemes have the desired resolution properties and thus, coupled with a nonoscillatory limiter, are ideal candidates for the numerical solution of such flows. A class of compact-reconstruction weighted essentially non-oscillatory CRWENO schemes is presented in this paper where lower order compact stencils are identified at each interface and combined using the WENO weights. This yields a higher order compact scheme for smooth solu- tions with superior resolution and lower truncation errors, compared to the WENO schemes. Across discontinuities, the scheme reduces to a lower order nonoscillatory compact scheme by excluding stencils containing the discontinuity. The schemes are analyzed for scalar conservation laws in terms of accuracy, convergence, and computational expense, and extended to the Euler equations of fluid dynamics. The scalar reconstruction is applied to the conserved and characteristic variables. Nu- merical test cases are presented that show the benefits of these schemes over the traditional WENO schemes.

On the integrability of a generalized variable-coefficient Kadomtsev–Petviashvili equation

Abstract: By considering the inhomogeneities of media, a generalized variable-coefficient Kadomtsev?Petviashvili (vc-KP) equation is investigated, which can be used to describe many nonlinear phenomena in fluid dynamics and plasma physics. In this paper, we systematically investigate the complete integrability of the generalized vc-KP equation under an integrable constraint condition. With the aid of generalized Bell?s polynomials, its bilinear formalism, bilinear B?cklund transformations, Lax pairs and Darboux covariant Lax pairs are succinctly constructed, which can be reduced to the ones of several integrable equations such as KdV, cylindrical KdV, KP, cylindrical KP, generalized cylindrical KP, non-isospectral KP equations, etc. Moreover, the infinite conservation laws of the equation are found by using its Lax equations. All conserved densities and fluxes are given with explicit recursion formulas. Furthermore, an extra auxiliary variable is introduced to obtain the bilinear formalism, based on which, the soliton solutions and Riemann theta function periodic wave solutions are presented. The influence of inhomogeneity coefficients on solitonic structures and interaction properties are discussed for physical interest and possible applications by some graphic analysis. Finally, a limiting procedure is presented to analyze in detail the asymptotic behavior of the periodic waves and the relations between the periodic wave solutions and soliton solutions.

Effective action approach to cosmological perturbations in dark energy and modified gravity

Abstract: In light of upcoming observations modelling perturbations in dark energy and modified gravity models has become an important topic of research. We develop an effective action to construct the components of the perturbed dark energy momentum tensor which appears in the perturbed generalized gravitational field equations, ?G?? = 8?G?T??+?U?? for linearized perturbations. Our method does not require knowledge of the Lagrangian density of the dark sector to be provided, only its field content. The method is based on the fact that it is only necessary to specify the perturbed Lagrangian to quadratic order and couples this with the assumption of global statistical isotropy of spatial sections to show that the model can be specified completely in terms of a finite number of background dependent functions. We present our formalism in a coordinate independent fashion and provide explicit formulae for the perturbed conservation equation and the components of ?U?? for two explicit generic examples: (i) the dark sector does not contain extra fields, = (g??) and (ii) the dark sector contains a scalar field and its first derivative = (g??,,??). We discuss how the formalism can be applied to modified gravity models containing derivatives of the metric, curvature tensors, higher derivatives of the scalar fields and vector fields.

Gravitational radiation reaction along general orbits in the effective one-body formalism

Abstract: We derive the gravitational radiation-reaction force modifying the Effective One Body (EOB) description of the conservative dynamics of binary systems. Our result is applicable to general orbits (elliptic or hyperbolic) and keeps terms of fractional second post-Newtonian order (but does not include tail effects). Our derivation of radiation-reaction is based on a new way of requiring energy and angular momentum balance. We give several applications of our results, notably the value of the (minimal) “Schott” contribution to the energy, the radial component of the radiationreaction force, and the radiative contribution to the angle of scattering during hyperbolic encounters. We present also new results about the conservative relativistic dynamics of hyperbolic motions.

Pseudospin Symmetry in Single Particle Resonant States

Abstract: The pseudospin symmetry (PSS) is a relativistic dynamical symmetry connected with the small component of the Dirac spinor The origin of PSS in single particle bound states in atomic nuclei has been revealed and studied extensively By examining the zeros of Jost functions corresponding to the small components of Dirac wave functions and phase shifts of continuum states, we show that the PSS in single particle resonant states in nuclei is conserved when the attractive scalar and repulsive vector potentials have the same magnitude but opposite sign The exact conservation and the breaking of the PSS are illustrated for single particle resonances in spherical square-well and Woods-Saxon potentials

An Alternative Formulation of Finite Difference Weighted ENO Schemes with Lax--Wendroff Time Discretization for Conservation Laws

Abstract: We develop an alternative formulation of conservative finite difference weighted essentially nonoscillatory (WENO) schemes to solve conservation laws. In this formulation, the WENO interpolation of the solution and its derivatives are used to directly construct the numerical flux, instead of the usual practice of reconstructing the flux functions. Even though this formulation is more expensive than the standard formulation, it does have several advantages. The first advantage is that arbitrary monotone fluxes can be used in this framework, while the traditional practice of reconstructing flux functions can be applied only to smooth flux splitting. The second advantage, which is fully explored in this paper, is that a narrower effective stencil is used compared with previous high order finite difference WENO schemes based on the reconstruction of flux functions, with a Lax--Wendroff time discretization. We will describe the scheme formulation and present numerical tests for one- and two-dimensional scalar ...

The cauchy problem for conservation laws with a multiplicative stochastic perturbation

Abstract: We study the Cauchy problem for multi-dimensional nonlinear conservation laws with multiplicative stochastic perturbation. Using the concept of measure-valued solutions and Kruzhkov's entropy formulation, the existence and uniqueness of an entropy solution is established.

Weak–Strong Uniqueness of Dissipative Measure-Valued Solutions for Polyconvex Elastodynamics

Abstract: For the equations of elastodynamics with polyconvex stored energy, and some related simpler systems, we define a notion of a dissipative measure-valued solution and show that such a solution agrees with a classical solution with the same initial data, when such a classical solution exists. As an application of the method we give a short proof of strong convergence in the continuum limit of a lattice approximation of one dimensional elastodynamics in the presence of a classical solution. Also, for a system of conservation laws endowed with a positive and convex entropy, we show that dissipative measure-valued solutions attain their initial data in a strong sense after time averaging.

The WAY theorem and the quantum resource theory of asymmetry

Abstract: The WAY theorem establishes an important constraint that conservation laws impose on quantum mechanical measurements. We formulate the WAY theorem in the broader context of resource theories, where one is constrained to a subset of quantum mechanical operations described by a symmetry group. Establishing connections with the theory of quantum state discrimination we obtain optimal unitaries describing the measurement of arbitrary observables, explain how prior information can permit perfect measurements that circumvent the WAY constraint, and provide a framework that establishes a natural ordering on measurement apparatuses through a decomposition into asymmetry and charge subsystems.

Singular solutions of a fully nonlinear 2 × 2 system of conservation laws

Abstract: Existence and admissibility of δ-shock solutions is discussed for the non-convex strictly hyper- bolic system of equations ∂tu + ∂x( 1 (u 2 + v 2 )) = 0, ∂tv + ∂x(v(u − 1)) = 0. The system is fully nonlinear, i.e. it is nonlinear with respect to both unknowns, and it does not admit the classical Lax-admissible solution for certain Riemann problems. By introducing complex-valued cor- rections in the framework of the weak asymptotic method, we show that a compressive δ-shock solution resolves such Riemann problems. By letting the approximation parameter tend to zero, the corrections become real valued, and the solutions can be seen to fit into the framework of weak singular solutions defined by Danilov and Shelkovich. Indeed, in this context, we can show that every 2 × 2 system of conservation laws admits δ-shock solutions.

Nonlinear stability of mKdV breathers

Abstract: Breather solutions of the modified Korteweg-de Vries equation are shown to be globally stable in a natural H^2 topology. Our proof introduces a new Lyapunov functional, at the H^2 level, which allows to describe the dynamics of small perturbations, including oscillations induced by the periodicity of the solution, as well as a direct control of the corresponding instability modes. In particular, degenerate directions are controlled using low-regularity conservation laws.