In this paper, we investigate an original way to deal with the problems generated by the limitation process of high-order finite volume methods based on polynomial reconstructions. Multi-dimensional ...Optimal Order Detection (MOOD) breaks away from classical limitations employed in high-order methods. The proposed method consists of detecting problematic situations after each time update of the solution and of reducing the local polynomial degree before recomputing the solution. As multi-dimensional MUSCL methods, the concept is simple and independent of mesh structure. Moreover MOOD is able to take physical constraints such as density and pressure positivity into account through an “a posteriori” detection. Numerical results on classical and demanding test cases for advection and Euler system are presented on quadrangular meshes to support the promising potential of this approach.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UL, UM, UPCLJ, UPUK
Finite difference WENO schemes have established themselves as very worthy performers for entire classes of applications that involve hyperbolic conservation laws. In this paper we report on two major ...advances that make finite difference WENO schemes more efficient.
The first advance consists of realizing that WENO schemes require us to carry out stencil operations very efficiently. In this paper we show that the reconstructed polynomials for any one-dimensional stencil can be expressed most efficiently and economically in Legendre polynomials. By using Legendre basis, we show that the reconstruction polynomials and their corresponding smoothness indicators can be written very compactly. The smoothness indicators are written as a sum of perfect squares. Since this is a computationally expensive step, the efficiency of finite difference WENO schemes is enhanced by the innovation which is reported here.
The second advance consists of realizing that one can make a non-linear hybridization between a large, centered, very high accuracy stencil and a lower order WENO scheme that is nevertheless very stable and capable of capturing physically meaningful extrema. This yields a class of adaptive order WENO schemes, which we call WENO-AO (for adaptive order). Thus we arrive at a WENO-AO(5,3) scheme that is at best fifth order accurate by virtue of its centered stencil with five zones and at worst third order accurate by virtue of being non-linearly hybridized with an r=3 CWENO scheme. The process can be extended to arrive at a WENO-AO(7,3) scheme that is at best seventh order accurate by virtue of its centered stencil with seven zones and at worst third order accurate. We then recursively combine the above two schemes to arrive at a WENO-AO(7,5,3) scheme which can achieve seventh order accuracy when that is possible; graciously drop down to fifth order accuracy when that is the best one can do; and also operate stably with an r=3 CWENO scheme when that is the only thing that one can do. Schemes with ninth order of accuracy are also presented.
Several accuracy tests and several stringent test problems are presented to demonstrate that the method works very well.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UL, UM, UPCLJ, UPUK
Exceptional points in optics and photonics Miri, Mohammad-Ali; Alù, Andrea
Science (American Association for the Advancement of Science),
01/2019, Volume:
363, Issue:
6422
Journal Article
Peer reviewed
Exceptional points are branch point singularities in the parameter space of a system at which two or more eigenvalues, and their corresponding eigenvectors, coalesce and become degenerate. Such ...peculiar degeneracies are distinct features of non-Hermitian systems, which do not obey conservation laws because they exchange energy with the surrounding environment. Non-Hermiticity has been of great interest in recent years, particularly in connection with the quantum mechanical notion of parity-time symmetry, after the realization that Hamiltonians satisfying this special symmetry can exhibit entirely real spectra. These concepts have become of particular interest in photonics because optical gain and loss can be integrated and controlled with high resolution in nanoscale structures, realizing an ideal playground for non-Hermitian physics, parity-time symmetry, and exceptional points. As we control dissipation and amplification in a nanophotonic system, the emergence of exceptional point singularities dramatically alters their overall response, leading to a range of exotic optical functionalities associated with abrupt phase transitions in the eigenvalue spectrum. These concepts enable ultrasensitive measurements, superior manipulation of the modal content of multimode lasers, and adiabatic control of topological energy transfer for mode and polarization conversion. Non-Hermitian degeneracies have also been exploited in exotic laser systems, new nonlinear optics schemes, and exotic scattering features in open systems. Here we review the opportunities offered by exceptional point physics in photonics, discuss recent developments in theoretical and experimental research based on photonic exceptional points, and examine future opportunities in this area from basic science to applied technology.
We propose a theory that describes quantitatively the (in)stability of fully many-body localization (MBL) systems due to ergodic, i.e., delocalized, grains, that can be, for example, due to disorder ...fluctuations. The theory is based on the ETH hypothesis and elementary notions of perturbation theory. The main idea is that we assume as much chaoticity as is consistent with conservation laws. The theory describes correctly-even without relying on the theory of local integrals of motion (LIOM)-the MBL phase in one dimension at strong disorder. It yields an explicit and quantitative picture of the spatial boundary between localized and ergodic systems. We provide numerical evidence for this picture. When the theory is taken to its extreme logical consequences, it predicts that the MBL phase is destabilised in the long time limit whenever (1) interactions decay slower than exponentially in d=1 and (2) always in d>1. Finer numerics is required to assess these predictions.
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Weak perturbations can drive an interacting many-particle system far from its initial equilibrium state if one is able to pump into degrees of freedom approximately protected by conservation laws. ...This concept has for example been used to realize Bose-Einstein condensates of photons, magnons and excitons. Integrable quantum systems, like the one-dimensional Heisenberg model, are characterized by an infinite set of conservation laws. Here, we develop a theory of weakly driven integrable systems and show that pumping can induce large spin or heat currents even in the presence of integrability breaking perturbations, since it activates local and quasi-local approximate conserved quantities. The resulting steady state is qualitatively captured by a truncated generalized Gibbs ensemble with Lagrange parameters that depend on the structure but not on the overall amplitude of perturbations nor the initial state. We suggest to use spin-chain materials driven by terahertz radiation to realize integrability-based spin and heat pumps.
This paper shows that the discontinuous Galerkin collocation spectral element method with Gauss--Lobatto points (DGSEM-GL) satisfies the discrete summation-by-parts (SBP) property and can thus be ...classified as an SBP-SAT (simultaneous approximation term) scheme with a diagonal norm operator. In the same way, SBP-SAT finite difference schemes can be interpreted as discontinuous Galerkin-type methods with a corresponding weak formulation based on an inner-product formulation common in the finite element community. This relation allows the use of matrix-vector notation (common in the SBP-SAT finite difference community) to show discrete conservation for the split operator formulation of scalar nonlinear conservation laws for DGSEM-GL and diagonal norm SBP-SAT. Based on this result, a skew-symmetric energy stable discretely conservative DGSEM-GL formulation (applicable to general diagonal norm SBP-SAT schemes) for the nonlinear Burgers equation is constructed. PUBLICATION ABSTRACT
We consider two physically and mathematically distinct regularization mechanisms of scalar hyperbolic conservation laws. When the flux is convex, the combination of diffusion and dispersion is known ...to give rise to monotonic and oscillatory traveling waves that approximate shock waves. The zero-diffusion limits of these traveling waves are dynamically expanding dispersive shock waves (DSWs). A richer set of wave solutions can be found when the flux is nonconvex. This review compares the structure of solutions of Riemann problems for a conservation law with nonconvex, cubic flux regularized by two different mechanisms: (1) dispersion in the modified Korteweg—de Vries (mKdV) equation; and (2) a combination of diffusion and dispersion in the mKdV—Burgers equation. In the first case, the possible dynamics involve two qualitatively different types of DSWs, rarefaction waves (RWs), and kinks (monotonie fronts). In the second case, in addition to RWs, there are traveling wave solutions approximating both classical (Lax) and nonclassical (undercompressive) shock waves. Despite the singular nature of the zero-diffusion limit and rather differing analytical approaches employed in the descriptions of dispersive and diffusive-dispersive regularization, the resulting comparison of the two cases reveals a number of striking parallels. In contrast to the case of convex flux, the mKdVB to mKdV mapping of Riemann problem solutions is not one-to—one, The mKdV kink solution is identified as an undercompressive DSW. Other prominent features, such as shock-rarefactions, also find their purely dispersive counterparts involving special contact DSWs, which exhibit features analogous to contact discontinuities. This review describes an important link between two major areas of applied mathematics: hyperbolic conservation laws and nonlinear dispersive waves.
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Operationally accessible entanglement in bipartite systems of indistinguishable particles could be reduced due to restrictions on the allowed local operations as a result of particle number ...conservation. In order to quantify this effect, Wiseman and Vaccaro Phys. Rev. Lett. 91, 097902 (2003)PRLTAO0031-900710.1103/PhysRevLett.91.097902 introduced an operational measure of the von Neumann entanglement entropy. Motivated by advances in measuring Rényi entropies in quantum many-body systems subject to conservation laws, we derive a generalization of the operationally accessible entanglement that is both computationally and experimentally measurable. Using the Widom theorem, we investigate its scaling with the size of a spatial subregion for free fermions and find a logarithmically violated area law scaling, similar to the spatial entanglement entropy, with at most a double-log leading-order correction. A modification of the correlation matrix method confirms our findings in systems of up to 10^{5} particles.
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A new technique for constructing conservation laws for fractional differential equations not having a Lagrangian is proposed. The technique is based on the methods of Lie group analysis and employs ...the concept of nonlinear self-adjointness which is enhanced to the certain class of fractional evolution equations. The proposed approach is demonstrated on subdiffusion and diffusion-wave equations with the Riemann–Liouville and Caputo time-fractional derivatives. It is shown that these equations are nonlinearly self-adjoint, and therefore, desired conservation laws can be calculated using the appropriate formal Lagrangians. The explicit forms of fractional generalizations of the Noether operators are also proposed for the equations with the Riemann–Liouville and Caputo time-fractional derivatives of order
α
∈
(
0
,
2
)
. Using these operators and formal Lagrangians, new conservation laws are constructed for the linear and nonlinear time-fractional subdiffusion and diffusion-wave equations by their Lie point symmetries.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
We propose a conservative physics-informed neural network (cPINN) on discrete domains for nonlinear conservation laws. Here, the term discrete domain represents the discrete sub-domains obtained ...after division of the computational domain, where PINN is applied and the conservation property of cPINN is obtained by enforcing the flux continuity in the strong form along the sub-domain interfaces. In case of hyperbolic conservation laws, the convective flux contributes at the interfaces, whereas in case of viscous conservation laws, both convective and diffusive fluxes contribute. Apart from the flux continuity condition, an average solution (given by two different neural networks) is also enforced at the common interface between two sub-domains. One can also employ a deep neural network in the domain, where the solution may have complex structure, whereas a shallow neural network can be used in the sub-domains with relatively simple and smooth solutions. Another advantage of the proposed method is the additional freedom it gives in terms of the choice of optimization algorithm and the various training parameters like residual points, activation function, width and depth of the network etc. Various forms of errors involved in cPINN such as optimization, generalization and approximation errors and their sources are discussed briefly. In cPINN, locally adaptive activation functions are used, hence training the model faster compared to its fixed counterparts. Both, forward and inverse problems are solved using the proposed method. Various test cases ranging from scalar nonlinear conservation laws like Burgers, Korteweg–de Vries (KdV) equations to systems of conservation laws, like compressible Euler equations are solved. The lid-driven cavity test case governed by incompressible Navier–Stokes equation is also solved and the results are compared against a benchmark solution. The proposed method enjoys the property of domain decomposition with separate neural networks in each sub-domain, and it efficiently lends itself to parallelized computation, where each sub-domain can be assigned to a different computational node.
•Domain decomposition technique is proposed for PINNs with tailored neural network in each sub-domain for solving conservation laws.•Due to presence of multiple neural networks, the representation capacity of the proposed cPINN method increases.•Based on prior knowledge of the solution regularity in each sub-domain, the hyper-parameter set of corresponding PINN can be properly adjusted.•The partial independence of individual PINNs in decomposed domains can be further employed to implement cPINN in a parallelized algorithm.
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