Higher Spatial Derivatives Research Articles

A Comparison of Hořava–Lifshitz Gravity and Einstein Gravity through the Gravitational Deflection of Massive Body around Black Holes

Hořava has proposed a renormalizable gravity theory with higher spatial derivatives in four dimensions. This theory may be regarded as a UV complete candidate for general relativity. After the proposal of this theory, Kehagias and Sfetsos have found a new asymptotically flat black hole solution in Hořava–Lifshitz gravity. In recent times, a new test of gravity theory is suggested that assumes the deflection of the massive body around a black hole. In this paper, we will study the effect of the Hořava–Lifshitz parameters on the black hole deflection angle and emphasize those features that permit a comparison of Hořava–Lifshitz to Einstein gravity.

Superluminal propagation from IR physics

One may believe that front velocities of waves in a given theory coincide with the UV limit of phase velocities for any dispersion relations. This implies that IR physics is irrelevant to the discussion of propagation speed of waves. We first consider a theory that contains higher spatial derivatives in the wave equation and prove that front velocities coincide with the UV limit of phase velocities, at least, if parity is conserved. However, we also show that front velocities do not coincide with the UV limit of phase velocities in general dispersion relations. We explicitly give several examples in which front velocities are superluminal owing to an IR or intermediate energy scale property of dispersion relations even if the UV limit of phase velocities is luminal. Our finding conveys the important caution that not only UV physics but also IR physics can be significant to superluminality.

Effect of the Coefficient on the Performance of A Two-Layer Boussinesq-Type Model

The coefficients embodied in a Boussinesq-type model are very important since they are determined to optimize the linear and nonlinear properties. In most conventional Boussinesq-type models, these coefficients are assigned the specific values. As for the multi-layer Boussinesq-type models with the inclusion of the vertical velocity, however, the effect of the different values of these coefficients on linear and nonlinear performances has never been investigated yet. The present study focuses on a two-layer Boussinesq-type model with the highest spatial derivatives being 2 and theoretically and numerically examines the effect of the coefficient a on model performance. Theoretical analysis show that different values for α (0.13⩽α⩽0.25) do not have great effects on the high accuracy of the linear shoaling, linear phase celerity and even third-order nonlinearity for water depth range of 0<kh⩽10 (k is wave number and h is water depth). The corresponding errors using different α values are restricted within 0.1%, 0.1% and 1% for the linear shoaling amplitude, dispersion and nonlinear harmonics, respectively. Numerical tests including regular wave shoaling over mildly varying slope from deep to shallow water, regular wave propagation over submerged bar, bichromatic wave group and focusing wave propagation over deep water are conducted. The comparison between numerical results using different values of α, experimental data and analytical solutions confirm the theoretical analysis. The flexibility and consistency of the two-layer Boussinesq-type model is therefore demonstrated theoretically and numerically.

A multi-layer Boussinesq-type model with second-order spatial derivatives: Theoretical analysis and numerical implementation

The accuracy of the linear and nonlinear properties embodied in multi-layer Boussinesq-type models with the highest spatial derivative n being 2, as proposed by Liu et al. (2018), is theoretically and numerically investigated in this paper. Theoretical analysis shows that the four-layer model has the highest accuracy and is applicable up to kh = 179.3 (where k is the wavenumber, and h is a typical water depth) in phase celerity at a 1% tolerance error, kh = 69.7 in the linear velocity components and kh = 141.8 in linear shoaling amplitude. At the same tolerance error, the super- and sub-harmonic transfer functions are accurate up to kh = 138.5 and 80, respectively, and the third-order harmonics and amplitude dispersion are accurate up to kh = 127.2. A high accuracy of the linear velocity profiles is also achieved and is approximately kh = 4.5, 14.9 and 69.7 for the two-layer, three-layer and four-layer models, respectively. Vertical two dimensional (2D) numerical models are established with a composite fourth-order Adams-Bashforth-Moulton scheme in time integration. Numerical simulations, including linear shoaling of regular waves over a mild slope, nonlinear regular wave evolution over a submerged breakwater and focusing wave group evolution over a constant water depth, are carried out. The computed results are in reasonable agreement with the experimental data. Furthermore, the CPU times for the two-layer model with n = 2 and the two-layer model with n = 3 are compared, and the numerical efficiency of the two-layer model with n = 2 increases by approximately 1/3.

Mixed-Coordinate ANCF Rectangular Plate Finite Element

Higher order finite elements (FEs) based on the absolute nodal coordinate formulation (ANCF) may require the use of curvature vectors as nodal coordinates. The curvature vectors, however, can be difficult to define at the reference configuration, making such higher order ANCF FEs less attractive to use. It is the objective of this investigation to use the concept of the mixed-coordinate ANCF FEs to ensure that the gradient vectors are the highest spatial derivatives in the element nodal coordinate vector regardless of the order of the interpolating polynomials used. This concept is used to convert the curvature vectors to nodes, called position nodes, which have only position coordinates. These new position nodes can be defined at a preprocessing stage, leading to two different sets of nodes: one set of nodes has position and gradient coordinates, while the second set of nodes has position coordinates only. The new position nodes can be used to obtain better distribution of the forces, including contact forces. Higher degree of continuity, including curvature continuity, can still be achieved at the element interface by using, at a preprocessing stage, linear algebraic equations that can reduce significantly the model dimension and ensure higher degree of smoothness. The procedure proposed in this investigation also allows for the formulation of mechanical joints at arbitrary points and nodes using linear algebraic constraint equations. The difficulties that arise when formulating these joint constraints using B-spline and NURBS (Nonuniform Rational B-Splines) representations are discussed. In order to explain the concepts introduced in this paper, low and high order ANCF thin plate elements are used. For the high order thin plate element, the curvature vectors at the interface nodes are converted to internal nodes with position coordinates only, leading to a mixed-coordinate ANCF thin plate element. This element preserves the desirable ANCF features including a constant mass matrix and zero Coriolis and centrifugal forces. Kirchhoff plate theory is used to formulate the element elastic forces. The equations of motion of the structure are formulated in terms of an independent set of structure coordinates. The resulting mass matrix associated with the independent coordinates remains constant. Numerical examples are presented in order to demonstrate the use of the mixed-coordinate ANCF thin plate element when the continuity constraints are imposed.

New Time Differencing Methods for Spectral Methods

A new semi-analytical time differencing is applied to spectral methods for partial differential equations which involve higher spatial derivatives. This is developed in Jung and Nguyen (J Sci Comput (2015) 63:355---373) based on the classical integrating factor (IF) and exponential time differencing (ETD) methods. The basic idea is approximating analytically the stiffness (fast part) by the so-called correctors (see 1.3 below) and numerically the non-stiffness (slow part) by the IF and ETD, etc. It turns out that rapid decay and rapid oscillatory modes in the spectral methods are well approximated by our corrector methods, which in turn provides better accuracy in the numerical schemes presented in the text. We investigate some nonlinear problems with a quadratic nonlinear term, which makes all Fourier modes interact with each other. We construct the correctors recursively to accurately capture the stiffness in the mode interactions. Polynomial or other types of nonlinear interactions can be tackled in a similar fashion.

Chern-Simons terms in Lifshitz-like quantum electrodynamics

In this work the generation of generalized Chern-Simons terms in three-dimensional quantum electrodynamics with high spatial derivatives is studied. We analyze the self-energy corrections to the gauge field propagator by considering an expansion of the corresponding amplitudes up to third order in the external momenta. The divergences of the corrections are determined and explicit forms for the Chern-Simons terms with high derivatives are obtained. Some unusual aspects of the calculation are stressed and the existence of a smooth isotropic limit is proved. The transversality of the anisotropic gauge propagator is also discussed.

Higher spatial derivative field theories

In this work, we employ renormalization group methods to study the general behavior of field theories possessing anisotropic scaling in the spacetime variables. The Lorentz symmetry breaking that accompanies these models are either soft, if no higher spatial derivative is present, or it may have a more complex structure if higher spatial derivatives are also included. Both situations are discussed in models with only scalar fields and also in models with fermions as a Yukawa-like model.

A High Order Spectral Volume Formulation for Solving Equations Containing Higher Spatial Derivative Terms II: Improving the Third Derivative Spatial Discretization Using the LDG2 Method

A High Order Spectral Volume Formulation for Solving Equations Containing Higher Spatial Derivative Terms II: Improving the Third Derivative Spatial Discretization Using the LDG2 Method

Emergent universe in Hor̆ava gravity

Recently Horava has proposed a non-relativistic renormalisable gravity theory with higher spatial derivatives in four dimensions which reduces to Einstein’s gravity at large distances with a non-vanishing cosmological constant but with improved UV behaviour. In this paper, we have considered the Friedman-Lemaitre-Robertson-Walker cosmological model in Horava gravity and the emergent scenario for all values of the spatial curvature k (=0,±1) has been studied. As a result, there are constraints on the parameters involved.

Accurate transfer maps for realistic beam-line elements: Straight elements

The behavior of orbits in charged-particle beam transport systems, including both linear and circular accelerators as well as final focus sections and spectrometers, can depend sensitively on nonlinear fringe-field and high-order-multipole effects in the various beam-line elements. The inclusion of these effects requires a detailed and realistic model of the interior and fringe fields, including their high spatial derivatives. A collection of surface fitting methods has been developed for extracting this information accurately from three-dimensional field data on a grid, as provided by various three-dimensional finite-element field codes. Based on these realistic field models, Lie or other methods may be used to compute accurate design orbits and accurate transfer maps about these orbits. Part I of this work presents a treatment of straight-axis magnetic elements, while part II will treat bending dipoles with large sagitta. An exactly soluble but numerically challenging model field is used to provide a rigorous collection of performance benchmarks.

Testing Hořava-Lifshitz gravity using thin accretion disk properties

Recently, a renormalizable gravity theory with higher spatial derivatives in four dimensions was proposed by Ho\ifmmode \check{r}\else \v{r}\fi{}ava. The theory reduces to Einstein gravity with a nonvanishing cosmological constant in IR, but it has improved UV behaviors. The spherically symmetric black hole solutions for an arbitrary cosmological constant, which represent the generalization of the standard Schwarzschild--(anti) de Sitter solution, have also been obtained for the Ho\ifmmode \check{r}\else \v{r}\fi{}ava-Lifshitz theory. The exact asymptotically flat Schwarzschild-type solution of the gravitational field equations in Ho\ifmmode \check{r}\else \v{r}\fi{}ava gravity contains a quadratic increasing term, as well as the square root of a fourth order polynomial in the radial coordinate, and it depends on one arbitrary integration constant. The IR-modified Ho\ifmmode \check{r}\else \v{r}\fi{}ava gravity seems to be consistent with the current observational data, but in order to test its viability more observational constraints are necessary. In the present paper we consider the possibility of observationally testing Ho\ifmmode \check{r}\else \v{r}\fi{}ava gravity by using the accretion disk properties around black holes. The energy flux, the temperature distribution, the emission spectrum, as well as the energy conversion efficiency are obtained, and compared to the standard general relativistic case. Particular signatures can appear in the electromagnetic spectrum, thus leading to the possibility of directly testing Ho\ifmmode \check{r}\else \v{r}\fi{}ava gravity models by using astrophysical observations of the emission spectra from accretion disks.

Using High‐Order Methods on Adaptively Refined Block‐Structured Meshes: Derivatives, Interpolations, and Filters

Block‐structured adaptive mesh refinement (AMR) is used for efficient resolution of partial differential equations (PDEs) solved on large computational domains by clustering mesh points only where required by large gradients. Previous work has indicated that fourth‐order convergence can be achieved on such meshes by using a suitable combination of high‐order derivative, interpolation, and filter stencils and can deliver significant computational savings over conventional second‐order methods at engineering error tolerances. In this paper, we explore the interactions between the errors introduced by the discretized operators for derivatives, interpolations, and filters. We develop general expressions for high‐order derivative, interpolation, and filter stencils applicable in multiple dimensions, using a Fourier approach, to facilitate high‐order block‐structured AMR implementations. These stencils are derived under the assumption that the fields that they are applied to are smooth. For a given derivative stencil (and thus a given order of accuracy), the necessary interpolation order is found to be dependent on the highest spatial derivative in the PDE being solved. This is demonstrated empirically, using one‐ and two‐dimensional model equations, by observing the increase in delivered accuracy as the order of accuracy of the interpolation stencil is increased. We also examine the empirically observed order of convergence, as the effective resolution of the mesh is increased by successively adding levels of refinement, with different orders of derivative, interpolation, and filter stencils. The procedure devised here is modular and its various pieces, i.e., the derivative, interpolation, and filter stencils, can be extended independently to higher orders and dimensions (in the case of interpolation stencils). The application of these methods often requires nontrivial logic, especially near domain boundaries; this logic, along with the stencils used in this study, are available as a freely downloadable software library.

High Spatial Resolution Satellite Imagery, DEM Derivatives, and Image Segmentation for the Detection of Mass Wasting Processes

An automated approach to identifying landslides using a combination of high-resolution satellite imagery and digital elevation derivatives is offered as an alternative to aerial photographic interpretation. Previous research has demonstrated that per pixel spectral response patterns are ineffective in discriminating mass movements. This technique utilizes image segmentation and digital elevation data in order to identify mass movements based not only on their reflectance but also on their shape properties and their geomorphic context. Dividing the classification by process into debris slides, debris flows, and rock slides makes the method far more useful than methods that group all mass movements together. A hierarchical classification scheme is utilized to eliminate areas that are not of interest and to identify areas where mass movements are probable. A supervised classification is then conducted using spectral, shape, and textural properties to identify failures that were greater than 1 ha in area. The resulting accuracy was 90 percent for debris slides, 60 percent for debris flows, and 80 percent for rock slides.

The generalized Dirichlet-to-Neumann map for certain nonlinear evolution PDEs

Let q(x,t) satisfy a nonlinear integrable evolution PDE whose highest spatial derivative is of order n. An initial boundary value problem on the half-line for such a PDE is at least linearly well-posed if one prescribes initial conditions, as well as N boundary conditions at x = 0, where for n even N equals n/2 and for n odd, depending on the sign of the highest derivative, N equals either n−1/2 or n+1/2. For example, for the nonlinear Schrödinger (NLS) and the sine-Gordon (sG), N = 1, while for the modified Korteweg-deVries (mKdV) N = 1 or N = 2 depending on the sign of the third derivative. Constructing the generalized Dirichlet-to-Neumann map means determining those boundary values at x = 0 that are not prescribed as boundary conditions in terms of the given initial and boundary conditions. A general methodology is presented that constructs this map in terms of the solution of a system of two nonlinear ODEs. This formulation implies that for the focusing NLS, for the sG, and for the two focusing versions of the mKdV, this map is global in time. It appears that this is the first time in the literature that such a characterization for nonlinear PDEs is explicitly described. It is also shown here that for particular choices of the boundary conditions the above map can be linearized. © 2005 Wiley Periodicals, Inc.

Instability and collapse of waveguides on the water surface under the ice cover

We consider long gravity-flexural waves on a surface of a perfect liquid of a finite depth under elastic ice-plate. Such waves of small but finite amplitude are governed by the generalized Kadomtsev–Petviashvili (KP) equation, which contains higher spatial derivatives. The generalized KP equation admits waveguide solutions, describing waves periodic in the direction of propagation and localized in the transverse direction. Waveguide represents a wave being the nonlinear product of instability of carrier monochromatic wave with respect to transverse perturbations. Instability of waveguides in its nonlinear stage is studied. For this purpose we use the alternative description via the Davey–Stewartson (DS) equations for slowly varying amplitudes of monochromatic waves. The DS equations are asymptotically equivalent to the initial generalized KP equation. Behaviour of perturbations is determined by values of wavenumber of the carrier wave. We find, in particular, that for some range of wavenumbers of the carrier wave the waveguide is subjected to the local collapse which differs from collapse of waveguides in the fluid of infinite depth.

A nonlinear response of the BiSrCaCuO single crystal in the microwave region

Third-harmonic microwave radiation of the BiSrCaCuO superconducting single crystal was studied. Two modes of microwave field-sample interactions were observed. In a weak field, a strong increase in the intensity of radiation after switching on a constant magnetic field, a hysteresis between opposite scan directions, and different harmonic amplitudes depending on the conditions of cooling (in the presence or absence of a magnetic field) were observed. These observations can be described by the generalized Ginzburg-Landau functional taking into account higher spatial derivatives of the order parameter. At a high intensity of incident waves, a magnetic field almost did not influence third-harmonic radiation, and, accordingly, hysteresis was absent. This is likely to be evidence that, at high powers, third-harmonic radiation arises as a result of generation of vortices under the action of a high-frequency magnetic field.

Nonlinear microwave response of Bi-Sr-Ca-Cu-O single crystal at microwave frequencies

The third harmonic generation by monocrystalline Bi-Sr-Ca-Cu-O under the influence of high microwave power at frequency 9 GHz has been investigated. Two modes of operation have been found. For comparatively weak driving forces we observed that dc magnetic field significantly increased amplitude of the third harmonic hysteresis when the dc magnetic field was changed, and different amplitude of the third harmonic for zero- and nonzero-field cooled states. All these peculiarities could be explained in the framework of phenomenological Ginzburg-Landau theory if one takes into account the higher spatial derivatives of the order parameter. At strong driving forces we found only weak influence of dc magnetic field without any hysteresis. It probably indicates that the harmonic generation for these powers is the result of creating and destroying vortices by a high-frequency field.

Performance comparison of convex-nonquadratic priors for Bayesian tomographic reconstruction

Bayesian methods have formed the core of the development of reconstruction algorithms for emission tomography. In particular, there has been considerable interest in edge-preserving prior models, which are associated with smoothing penalty functions that are nonquadratic functions of nearby pixel differences. In spite of several advantages of nonconvex prior models, their use in routine applications has been hindered by several factors, such as the computational expense due to the nonconvexity of penalty functions and the difficulty in the selection of hyperparameters. We note here that, by choosing a penalty function which is nonquadratic but is still convex, both the problem of nonconvexity involved in some nonquadratic priors and the edge-oversmoothing problem of conventional quadratic priors may be avoided. In this paper, we use a class of two-dimensional smoothing splines with low (first) and high (second) spatial derivatives applied to convex-nonquadratic (CNQ) penalty functions. To compare quantitative performance of our new priors, we use the quantitation of bias/variance and total squared error over noise trials. Our numerical results show that a linear combination of first- and second-order spatial derivatives applied to CNQ penalty functions improves reconstructions in terms of total squared error.