Despite the success of neural networks at solving concrete physics problems, their use as a general-purpose tool for scientific discovery is still in its infancy. Here, we approach this problem by ...modeling a neural network architecture after the human physical reasoning process, which has similarities to representation learning. This allows us to make progress towards the long-term goal of machine-assisted scientific discovery from experimental data without making prior assumptions about the system. We apply this method to toy examples and show that the network finds the physically relevant parameters, exploits conservation laws to make predictions, and can help to gain conceptual insights, e.g., Copernicus' conclusion that the solar system is heliocentric.
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We prove unexpected decay of the L2-distance from the solution u(t) of a hyperbolic scalar conservation law, to some convex, flow-invariant target sets.
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It is well known that semi-discrete high order discontinuous Galerkin (DG) methods satisfy cell entropy inequalities for the square entropy for both scalar conservation laws (Jiang and Shu (1994) 39) ...and symmetric hyperbolic systems (Hou and Liu (2007) 36), in any space dimension and for any triangulations. However, this property holds only for the square entropy and the integrations in the DG methods must be exact. It is significantly more difficult to design DG methods to satisfy entropy inequalities for a non-square convex entropy, and/or when the integration is approximated by a numerical quadrature. In this paper, we develop a unified framework for designing high order DG methods which will satisfy entropy inequalities for any given single convex entropy, through suitable numerical quadrature which is specific to this given entropy. Our framework applies from one-dimensional scalar cases all the way to multi-dimensional systems of conservation laws. For the one-dimensional case, our numerical quadrature is based on the methodology established in Carpenter et al. (2014) 5 and Gassner (2013) 19. The main ingredients are summation-by-parts (SBP) operators derived from Legendre Gauss–Lobatto quadrature, the entropy conservative flux within elements, and the entropy stable flux at element interfaces. We then generalize the scheme to two-dimensional triangular meshes by constructing SBP operators on triangles based on a special quadrature rule. A local discontinuous Galerkin (LDG) type treatment is also incorporated to achieve the generalization to convection–diffusion equations. Extensive numerical experiments are performed to validate the accuracy and shock capturing efficacy of these entropy stable DG methods.
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4.
Conservation laws shape dissipation Rao, Riccardo; Esposito, Massimiliano
New journal of physics,
02/2018, Volume:
20, Issue:
2
Journal Article
Peer reviewed
Open access
Starting from the most general formulation of stochastic thermodynamics-i.e. a thermodynamically consistent nonautonomous stochastic dynamics describing systems in contact with several reservoirs-we ...define a procedure to identify the conservative and the minimal set of nonconservative contributions in the entropy production. The former is expressed as the difference between changes caused by time-dependent drivings and a generalized potential difference. The latter is a sum over the minimal set of flux-force contributions controlling the dissipative flows across the system. When the system is initially prepared at equilibrium (e.g. by turning off drivings and forces), a finite-time detailed fluctuation theorem holds for the different contributions. Our approach relies on identifying the complete set of conserved quantities and can be viewed as the extension of the theory of generalized Gibbs ensembles to nonequilibrium situations.
In this paper a new simple fifth order weighted essentially non-oscillatory (WENO) scheme is presented in the finite difference framework for solving the hyperbolic conservation laws. The new WENO ...scheme is a convex combination of a fourth degree polynomial with two linear polynomials in a traditional WENO fashion. This new fifth order WENO scheme uses the same five-point information as the classical fifth order WENO scheme 14,20, could get less absolute truncation errors in L1 and L∞ norms, and obtain the same accuracy order in smooth region containing complicated numerical solution structures simultaneously escaping nonphysical oscillations adjacent strong shocks or contact discontinuities. The associated linear weights are artificially set to be any random positive numbers with the only requirement that their sum equals one. New nonlinear weights are proposed for the purpose of sustaining the optimal fifth order accuracy. The new WENO scheme has advantages over the classical WENO scheme 14,20 in its simplicity and easy extension to higher dimensions. Some benchmark numerical tests are performed to illustrate the capability of this new fifth order WENO scheme.
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•The Painlevé test has been conducted to reveal the Painlevé-integrability of a novel (2+1)-dimensional nonlinear model.•Non-resonant soliton solutions, Bäacklund transformation, Lax pair and ...infinitely many conservation laws have been derived.•The criterion for the linear superposition principle has been given, which can be used to generate the resonant solutions.
The (2+1)-dimensional Kadomtsev-Petviashvili type equations describe the nonlinear phenomena and characteristics in oceanography, fluid dynamics and shallow water. In the literature, a novel (2+1)-dimensional nonlinear model is proposed, and the localized wave interaction solutions are studied including lump-kink and lump-soliton types. Hereby, it is of further value to investigate the integrability characteristics of this model. In this paper, we firstly conduct the Painlevé analysis and find it fails to pass the Painlevé test due to a non-vanishing compatibility condition at the highest resonance level. Then we derive the soliton solutions and give the formula of the N-soliton solution, which is proved by means of the Hirota condition. The criterion for the linear superposition principle is also given to generate the resonant solutions. Bäcklund transformation, Lax pair and infinitely many conservation laws are derived through the Hirota bilinear method and Bell polynomial approach. As a result, we have a more overall understanding of the integrability characteristics of this novel (2+1)-dimensional nonlinear model.
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In this paper, a new type of high-order finite difference and finite volume multi-resolution weighted essentially non-oscillatory (WENO) schemes is presented for solving hyperbolic conservation laws. ...We only use the information defined on a hierarchy of nested central spatial stencils and do not introduce any equivalent multi-resolution representation. These new WENO schemes use the same large stencils as the classical WENO schemes in 25,45, could obtain the optimal order of accuracy in smooth regions, and could simultaneously suppress spurious oscillations near discontinuities. The linear weights of such WENO schemes can be any positive numbers on the condition that their sum equals one. This is the first time that a series of unequal-sized hierarchical central spatial stencils are used in designing high-order finite difference and finite volume WENO schemes. These new WENO schemes are simple to construct and can be easily implemented to arbitrary high order of accuracy and in higher dimensions. Benchmark examples are given to demonstrate the robustness and good performance of these new WENO schemes.
•A new class of high order finite difference and finite volume WENO schemes are constructed.•These schemes are based on the multi-resolution idea, and a series of unequal-sized hierarchical central spatial stencils.•These schemes can use arbitrary positive linear weights, and are easy to implement for one and multi-dimensions.•These schemes have a gradual degrading of accuracy near discontinuities.
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The ten-moment equations are considered as a first-order alternative of Navier-Stokes equations when the effect of heat transfer is negligible. This model takes the form of first-order hyperbolic ...conservation laws, which carry many numerical advantages. However, the applicability of this model is still limited due to the lack of appropriate turbulence models. Applying the Reynolds-averaging concept to the ten-moment model, a set of governing equations for turbulent flow can be obtained, which is referred to as the Reynolds-averaged ten-moment equations. The traditional turbulence models designed for the Reynolds-averaged Navier-Stokes (RANS) equations are not ideal for the Reynolds-averaged ten-moment equations, as the extra partial differential equations (PDEs) introduce second-order derivatives. These terms destroy the pure hyperbolic nature of the original system of equations, which consequently removes all numerical advantages of first-order systems. To maintain the first-order hyperbolic form, the desired turbulence model should remain in the same form. Recently, a hyperbolic-relaxation turbulence model has been proposed by the authors, which is developed by hyperbolizing Prandtl's one-equation model using a relaxation method known as the Chen-Levermore-Liu p-system. Unfortunately, developing a hyperbolic version of two-equation models using the same method is very difficult. This is because the diffusion coefficients of the two-equation models are more complicated than in the one-equation model. In this paper, another relaxation method, the Cattaneo-Vernotte approach, is used to develop the hyperbolic-relaxation form of classical two-equation models. The solution of the resulting equations exhibits dispersive wave behaviour. To study this feature, a dispersion analysis of the Reynolds-averaged ten-moment equations with the new turbulence models is presented. Several numerical experiments are studied to investigate the effect of the relaxation parameters. The derived turbulence models are then coupled to the Reynolds-averaged ten-moment equation and further validated by solving a canonical two-dimensional turbulent plane mixing-layer problem, planar free-jet problem, and circular free-jet problem.
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An alternative formulation of conservative weighted essentially non-oscillatory (WENO) finite difference scheme with the classical WENO-JS weights (Jiang et al. (2013) 6) has been successfully used ...for solving hyperbolic conservation laws. However, it fails to achieve the optimal order of accuracy at the critical points of a smooth function. Here, we demonstrate that the WENO-Z weights (Borges et al. (2008) 1) should be employed to recover the optimal order of accuracy at the critical points. Several one- and two-dimensional benchmark problems show the improved performance in terms of accuracy, resolution and shock capturing.
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Entropy solutions have been widely accepted as the suitable solution framework for systems of conservation laws in several space dimensions. However, recent results in De Lellis and Székelyhidi Jr ...(Ann Math 170(3):1417–1436,
2009
) and Chiodaroli et al. (
2013
) have demonstrated that entropy solutions may not be unique. In this paper, we present numerical evidence that state-of-the-art numerical schemes
need not
converge to an entropy solution of systems of conservation laws as the mesh is refined. Combining these two facts, we argue that entropy solutions may not be suitable as a solution framework for systems of conservation laws, particularly in several space dimensions. We advocate
entropy measure-valued solutions
, first proposed by DiPerna, as the appropriate solution paradigm for systems of conservation laws. To this end, we present a detailed numerical procedure which constructs stable approximations to entropy measure-valued solutions, and provide sufficient conditions that guarantee that these approximations converge to an entropy measure-valued solution as the mesh is refined, thus providing a viable numerical framework for systems of conservation laws in several space dimensions. A large number of numerical experiments that illustrate the proposed paradigm are presented and are utilized to examine several interesting properties of the computed entropy measure-valued solutions.
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