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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DOBA, EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, IZUM, KILJ, KISLJ, MFDPS, NLZOH, NUK, OBVAL, OILJ, PILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UILJ, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
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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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UL, UM, UPCLJ, UPUK, ZRSKP
We construct a solvable deformation of two-dimensional theories with (2, 2) supersymmetry using an irrelevant operator which is a bilinear in the supercurrents. This supercurrent-squared operator is ...manifestly supersymmetric and equivalent to TT¯ after using conservation laws. As illustrative examples, we deform theories involving a single (2, 2) chiral superfield. We show that the deformed free theory is on-shell equivalent to the (2, 2) Nambu-Goto action. At the classical level, models with a superpotential exhibit more surprising behavior: the deformed theory exhibits poles in the physical potential which modify the vacuum structure. This suggests that irrelevant deformations of TT¯ type might also affect infrared physics.
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We develop a mesoscopic lattice Boltzmann model for liquid-vapor phase transition by handling the microscopic molecular interaction. The short-range molecular interaction is incorporated by ...recovering an equation of state for dense gases, and the long-range molecular interaction is mimicked by introducing a pairwise interaction force. Double distribution functions are employed, with the density distribution function for the mass and momentum conservation laws and an innovative total kinetic energy distribution function for the energy conservation law. The recovered mesomacroscopic governing equations are fully consistent with kinetic theory, and thermodynamic consistency is naturally satisfied.
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We study conservation properties of Galerkin methods for the incompressible Navier–Stokes equations, without the divergence constraint strongly enforced. In typical discretizations such as the mixed ...finite element method, the conservation of mass is enforced only weakly, and this leads to discrete solutions which may not conserve energy, momentum, angular momentum, helicity, or vorticity, even though the physics of the Navier–Stokes equations dictate that they should. We aim in this work to construct discrete formulations that conserve as many physical laws as possible without utilizing a strong enforcement of the divergence constraint, and doing so leads us to a new formulation that conserves each of energy, momentum, angular momentum, enstrophy in 2D, helicity and vorticity (for reference, the usual convective formulation does not conserve most of these quantities). Several numerical experiments are performed, which verify the theory and test the new formulation.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UL, UM, UPCLJ, UPUK, ZRSKP
In this paper, a new type of high-order finite volume and finite difference multi-resolution Hermite weighted essentially non-oscillatory (HWENO) schemes are designed for solving hyperbolic ...conservation laws on structured meshes. Here we only use the information defined on a hierarchy of nested central spatial stencils but do not introduce any equivalent multi-resolution representation, the terminology of multi-resolution HWENO follows that of the multi-resolution WENO schemes (Zhu and Shu, 2018) 29. The main idea of our spatial reconstruction is derived from the original HWENO schemes (Qiu and Shu, 2004) 19, in which both the function and its first-order derivative values are evolved in time and used in the reconstruction. Our HWENO schemes use the same large stencils as the classical HWENO schemes which are narrower than the stencils of the classical WENO schemes for the same order of accuracy. Only the function values need to be reconstructed by our HWENO schemes, the first-order derivative values are obtained from the high-order linear polynomials directly. Furthermore, the linear weights of such HWENO schemes can be any positive numbers as long as their sum equals one, and there is no need to do any modification or positivity-preserving flux limiting in our numerical experiments. Extensive benchmark examples are performed to illustrate the robustness and good performance of such finite volume and finite difference HWENO schemes.
•High-order multi-resolution Hermite weighted essentially non-oscillatory (HWENO) schemes are designed.•Both finite volume and finite difference schemes are presented.•The new methods are robust, there are no need any additional limiter.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
We analyze the "higher rank" gauge theories that capture some of the phenomenology of the fracton order. It is shown that these theories lose gauge invariance when an arbitrarily weak and smooth ...curvature is introduced. We propose a resolution to this problem by introducing a theory invariant under area-preserving diffeomorphisms, which reduce to the higher rank gauge transformations upon linearization around a flat background. The proposed theory is geometric in nature and is interpreted as a theory of chiral topological elasticity. This theory exhibits some of the fracton phenomenology. We explore the conservation laws, topological excitations, linear response, various kinematical constraints, and canonical structure of the theory. Finally, we emphasize that the very structure of Riemann-Cartan geometry, which we use to formulate the theory, encodes some of the fracton phenomenology, suggesting that the fracton order itself is geometric in nature.
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Using underlying invariance/symmetry properties and related/associated conservation laws, we investigate some 'high' order nonlinear equations. The multiplier method is mainly used to construct ...conserved vectors for these equations. When the partial differential equations are reduced to the nonlinear ordinary differential equation (NLODE), exact solutions for the ODEs are constructed and graphical representations of the resulting solutions are provided. In some cases, the solutions obtained are the Jacobi elliptic cosine function and the solitary wave solutions. We study the third-order 'equal width equation' followed by a new fourth-order nonlinear partial differential equation (NLPDE), which was recently established in the literature and, finally, the Korteweg-de Vries (KdV) equation having three dispersion sources.
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Neural networks can emulate nonlinear physical systems with high accuracy, yet they may produce physically inconsistent results when violating fundamental constraints. Here, we introduce a systematic ...way of enforcing nonlinear analytic constraints in neural networks via constraints in the architecture or the loss function. Applied to convective processes for climate modeling, architectural constraints enforce conservation laws to within machine precision without degrading performance. Enforcing constraints also reduces errors in the subsets of the outputs most impacted by the constraints.
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