Fourth Derivative Terms Research Articles

Modified Fourth Derivative Block Method and its direct applications to third-order initial value problems

A theoretical order eight Modified Fourth Derivative four-step block method (MFDFBM) has been derived, analysed and numerically applied to solve multiple problems originating from Fluid Dynamics, engineering and other sciences. The MFDFBM was derived by applying collocation and interpolation techniques to a power series approximation. Further introducing fourth derivative terms at each of the collocating points yields a block method with an improved order of accuracy. It was observed that the order of the block method increases with the number of fourth derivative terms introduced into the integration interval. Numerical experiments are presented to test MFDFBM on numerical examples, including non-linear homogeneous thin film flow (NHTFF) problems and two non-linear initial value problems(IVPs). The experiments confirm the good impact of adding the fourth derivative terms, which help improve the order of accuracy of the derived MFDFBM, thereby minimising error and agreeing with analytical solution up to at least seven decimal places.

Revisiting the chiral effective action in holographic models

We obtain the pion decay constant and coefficients of fourth derivative terms in the chiral Lagrangian for massless quarks in the Witten-Sakai-Sugimoto model. We extract these quantities from the two-pion scattering amplitude, which we compute directly in the holographic dual through tree-level Witten diagrams. Identification of the low energy coefficients in the chiral action is subtle as their values will be shifted when the tower of massive vector bosons are integrated out. Indeed, by a direct comparison with the existing standard procedure of constructing the chiral action with radial modes in the gravity dual, we explicitly show that there are finite 't Hooft coupling corrections that have been missed. This suggests that past derivations of effective actions from holographic models may have to be revisited and future derivations more carefully considered.

Nonlocality of quantum matter corrections and cosmological constant running

Semiclassical loop corrections to the gravitational action include various terms with zero, two, and four derivatives of the metric as well as nonlocal form factors for these terms. Contributions to some of these terms could be confused with others on a specific metric background or for a particular gauge fixing. We present a critical analysis of the recent works where the tensor structure and the number of derivatives in the action of gravity were not properly taken into account. Taking these relevant aspects into account, we show that although some contributions owing to the quantum fluctuations of massive or massless scalar as well as fermion and vector fields may be attributed to the “running” of the cosmological constant, in reality they correspond to the fourth derivative terms of the action.

Kink dynamics in a nonlinear beam model

In this paper, we study the single kink and the kink-antikink collisions of a nonlinear beam equation bearing a fourth-derivative term. We numerically explore some of the key characteristics of the single kink both in its standing wave and in its traveling wave form. A point of emphasis is the study of kink-antikink collisions, exploring the critical velocity for single-bounce (and separation) and infinite-bounce (where the kink and antikink trap each other) windows. The relevant phenomenology turns out to be dramatically different than that of the corresponding nonlinear Klein-Gordon (i.e., ϕ4) model. Our computations show that for small initial velocities, the kink and antikink reflect nearly elastically without colliding. For an intermediate interval of velocities, the two waves trap each other, while for large speeds a single inelastic collision between them takes place. Lastly, we briefly touch upon the use of collective coordinates (CC) method and their predictions of the relevant phenomenology. When one degree of freedom is used in the CC approach, the results match well the numerical ones for small values of initial velocity. However, for bigger values of initial velocity, it is inferred that more degrees of freedom need to be self-consistently included in order to capture the collision phenomenology.

Constraints on Low-Energy Effective Theories from Weak Cosmic Censorship.

We examine the weak cosmic censorship conjecture (WCCC) for the extremal charged black hole in possible generalizations of Einstein-Maxwell theory due to the high-order corrections, up to fourth-derivative terms. Our derivation is based on Wald's gedanken experiment to destroy an extremal black hole. We find that the WCCC no longer holds for all possible generalizations. Thus, the WCCC can serve as a new constraint to the high-order effective field theories. However, our constraint is independent of photon's self-interactions so that precision measurement of quantum electrodynamics cannot constrain the WCCC. For higher-dimension operators induced by the one-loop correction for the minimally coupled spinor and scalar to gravity, our constraint is satisfied.

Well-posedness and Ill-posedness for Linear Fifth-Order Dispersive Equations in the Presence of Backwards Diffusion

Fifth-order dispersive equations arise in the context of higher-order models for phenomena such as water waves. For fifth-order variable-coefficient linear dispersive equations, we provide conditions under which the intitial value problem is either well-posed or ill-posed. For well-posedness, a balance must be struck between the leading-order dispersion and possible backwards diffusion from the fourth-derivative term. This generalizes work by the first author and Wright for third-order equations. In addition to inherent interest in fifth-order dispersive equations, this work is also motivated by a question from numerical analysis: finite difference schemes for third-order numerical equations can yield approximate solutions which effectively satisfy fifth-order equations. We find that such a fifth-order equation is well-posed if and only if the underlying third-order equation is ill-posed.

A numerical integration-based Kalman filter for moderately nonlinear systems

This paper introduces a computationally efficient data assimilation scheme based on Gaussian quadrature filtering that potentially outperforms current methods in data assimilation for moderately nonlinear systems. Moderately nonlinear systems, in this case, are systems with numerical models with small fourth and higher derivative terms. Gaussian quadrature filters are a family of filters that make simplifying Gaussian assumptions about filtering pdfs in order to numerically evaluate the integrals found in Bayesian data assimilation. These filters are differentiated by the varying quadrature rules to evaluate the arising integrals. The approach we present, denoted by Assumed Gaussian Reduced (AGR) filter, uses a reduced order version of the polynomial quadrature first proposed in Ito and Xiong [2000. Gaussian filters for nonlinear filtering problems. IEEE Trans. Automat. Control. 45, 910–927]. This quadrature uses the properties of Gaussian distributions to form an effectively higher order method increasing its efficiency. To construct the AGR filter, this quadrature is used to form a reduced order square-root filter, which will reduce computational costs and improve numerical robustness. For cases of sufficiently small fourth derivatives of the nonlinear model, we demonstrate that the AGR filter outperforms ensemble Kalman filters (EnKFs) for a Korteweg-de Vries model and a Boussinesq model.

Mean square responses of a viscoelastic Timoshenko cantilever beam with different damping mechanisms

In this paper, considering the moment of inertia and shear deformation, the partial differential equation sets governing the vibration of viscoelastic Timoshenko beam subjected to random excitations were derived. The excitation form is the concentrated force, and its random characteristic is the ideal white noise in the time domain. Two damping mechanisms, namely, the external viscous damping and the internal viscoelastic damping (Kelvin-Voigt model), are both taken into account simultaneously. Comparing with the existing literature, there are two improvements: on the one hand, considering the interaction of the rotational inertia and shear deformation, that is, there is a fourth derivative term for time in the partial differential equation sets of Timoshenko beam, while the Bresse-Timoshenko truncated model ignores the fourth derivative term of time. On the other hand, by the method of residue integral, the infinite integral in the mean response is transformed into the expression of the modal damping ratio and the natural frequency. The exact solution of the mean square response is obtained in the form of an infinite series finally. Numerical example is supplied, and the numerical results acquired verify the validity of the theoretical analysis.

Recent Progress in Fighting Ghosts in Quantum Gravity

We review some of the recent results which can be useful for better understanding of the problem of stability of vacuum and in general classical solutions in higher derivative quantum gravity. The fourth derivative terms in the purely gravitational vacuum sector are requested by renormalizability already in both semiclassical and complete quantum gravity theories. However, because of these terms, the spectrum of the theory has unphysical ghost states which jeopardize the stability of classical solutions. At the quantum level, ghosts violate unitarity, and thus ghosts look incompatible with the consistency of the theory. The “dominating” or “standard” approach is to treat higher derivative terms as small perturbations at low energies. Such an effective theory is supposed to glue with an unknown fundamental theory in the high energy limit. We argue that the perspectives for such a scenario are not clear, to say the least. On the other hand, recently, there was certain progress in understanding physical conditions which can make ghosts not offensive. We survey these results and discuss the properties of the unknown fundamental theory which can provide these conditions satisfied.

Improved numerical AC superposition method for electron energy distribution functions

An improved AC superposition method is proposed to obtain the electron energy distribution functions (EEPFs) by reducing the effect of the high order derivative terms. In the AC superposition method, a small AC voltage is applied to the current-voltage curve, and the EEPF is obtained by the second harmonic current which is proportional to the second derivative of the current. If the AC voltage is not small compared to the electron temperature, the EEPF can be distorted by high derivative terms that are not ignorable. Thus, the proposed method uses the DC component and the second harmonics of the current to reduce the influence of the high order derivative terms. The fourth derivative term is removed and the coefficients of the higher order derivative terms become ignorable. The EEPF by this method shows that the resolution of the low energy part in the EEPFs is greatly improved.

Construction, Analysis and Application of Coupled Compact Difference Scheme in Computational Acoustics and Fluid Flow Problems

AbstractIn the present work, a new type of coupled compact difference scheme has been proposed for the solution of computational acoustics and flow problems. The proposed scheme evaluates the first, the second and the fourth derivative terms simultaneously. Derived compact difference scheme has a significant spectral resolution and a physical dispersion relation preserving (DRP) ability over a considerable wavenumber range when a fourth order four stage Runge-Kutta scheme is used for the time integration. Central stencil has been used for the present numerical scheme to evaluate spatial derivative terms. Derived scheme has the capability of adding numerical diffusion adaptively to attenuate spurious high wavenumber oscillations responsible for numerical instabilities. The DRP nature of the proposed scheme across a wider wavenumber range provides accurate results for the model wave equations as well as computational acoustic problems. In addition to the attractive feature of adaptive diffusion, present scheme also helps to control spurious reflections from the domain boundaries and is projected as an alternative to the perfectly matched layer (PML) technique.

Gravitational waves and perspectives for quantum gravity

Understanding the role of higher derivatives is probably one of the most relevant questions in quantum gravity theory. Already at the semiclassical level, when gravity is a classical background for quantum matter fields, the action of gravity should include fourth derivative terms to provide renormalizability in the vacuum sector. The same situation holds in the quantum theory of metric. At the same time, including the fourth derivative terms means the presence of massive ghosts, which are gauge-independent massive states with negative kinetic energy. At both classical and quantum level such ghosts violate stability and hence the theory becomes inconsistent. Several approaches to solve this contradiction were invented and we are proposing one more, which looks simpler than those what were considered before. We explore the dynamics of the gravitational waves on the background of classical solutions and give certain arguments that massive ghosts produce instability only when they are present as physical particles. At least on the cosmological background one can observe that if the initial frequency of the metric perturbations is much smaller than the mass of the ghost, no instabilities are present.

A new high order space derivative discretization for 3D quasi-linear hyperbolic partial differential equations

In this paper, we propose a new high accuracy numerical method of O(k2+k2h2+h4) for the solution of three dimensional quasi-linear hyperbolic partial differential equations, where k>0 and h>0 are mesh sizes in time and space directions respectively. We mainly discretize the space derivative terms using fourth order approximation and time derivative term using second order approximation. We describe the derivation procedure in details and also discuss how our formulation is able to handle the wave equation in polar coordinates. The proposed method when applied to a linear hyperbolic equation is also shown to be unconditionally stable. The proposed method behaves like a fourth order method for a fixed value of (k/h2). Some examples and their numerical results are provided to justify the usefulness of the proposed method.

Some links between recent gradient thermo-elasto-plasticity theories and the thermomechanics of generalized continua

This work elaborates upon two robust models of gradient elasticity and gradient plasticity, and one gradient model of heat transfer, as originally advocated by the second author in the 1980’s. The objective is, after recalling the links between these models and existing generalized continuum theories as developed in the 1960’s and subsequently, to apply the same methodology to the case of diffusion with a view to establishing generalized transport equations. Aifantis double diffusivity and conductivity theory that provides generalized mass or heat transfer equations is compared to micromorphic-type hyper-temperature and micro-entropy proposals. The double temperature and the micromorphic thermal models are shown to lead to equations more general that Cattaneo’s. The sign of the coefficient of the second time-derivative of temperature is found to differ according to both approaches. The double temperature model contains a fourth space derivative term not present in the micromorphic models. Such generalized equations can be useful, for example, in the interpretation of recent femtosecond laser experiments on metals.

A New Derivation of the Perturbation Operator Used in MCNP

Abstract A new derivation of the perturbation operator used in the MCNP radiation transport code is presented, and equations are determined to evaluate the change in the response from a perturbation in macroscopic cross section due to the first, second, third, and fourth derivative terms, and the second-derivative cross term. The perturbation operator in MCNP currently can be used to determine the change in a response from the first and second order terms in Taylor expansion due to such a perturbation. The derivation presented here produces expressions that are equivalent to the expressions used in MCNP for the first and second derivative, but are simpler and are in a form that more readily reveals their physical significance. The perturbation operator is particularly useful for sensitivity and uncertainty analysis.

Ghosts, instabilities, and superluminal propagation in modified gravity models

We consider modified gravity models involving inverse powers of fourth order curvatureinvariants. Using these models’ equivalence to the theory of a scalar field coupled to alinear combination of the invariants, we investigate the properties of the propagatingmodes. Even in the case for which the fourth derivative terms in the field equations vanish,we find that the second derivative terms can give rise to ghosts, instabilities, andsuperluminal propagation speeds. We establish the conditions which the theories mustsatisfy in order to avoid these problems in Friedmann backgrounds, and show that the latetime attractor solutions generically exhibit superluminally propagating tensor or scalarmodes.

Phase transition to a square vortex lattice in type-II superconductors with fourfold anisotropy

We investigate the stability of the square vortex lattice which has been recently observed in experiments on the borocarbide family of superconductors. Taking into account the tetragonal symmetry of these systems, we add fourfold symmetric fourth-derivative terms to the Ginzburg-Landau(GL) free energy. At $H_{c2}$ these terms may be treated perturbatively to lowest order to locate the transition from a distorted hexagonal to a square vortex lattice. We also solve for this phase boundary numerically in the strongly type-II limit, finding large corrections to the lowest-order perturbative results. We calculate the relative fourfold $H_{c2}$ anisotropy for field in the $xy$ plane to be 4.5% at the temperature, $T_c^{\Box}$, where the transition occurs at $H_{c2}$ for field along the $z$ axis. This is to be compared to the 3.6% obtained in the perturbative calculation. Furthermore, we find that the phase boundary in the $H-T$ phase diagram has positive slope near $H_{c2}$.

Dynamic Renormalization Group Approach to the Kardar-Parisi-Zhang Equation and Its Results Derived

By removing a trick hidden in the original KPZ paper, a correction in the application of the dynamic renormalization group technique to the KPZ problem is proposed. It is stressed that a fourth derivative term plays an essential role in formulating a consistent scenario. In a one-loop approximation, α=2- z =8(4- d )/(48+4 d - d 2 ), is obtained for the roughness exponent α and the dynamic exponent z , which gives reasonable numbers. For 2< d <4, a smooth surface (α=0, z =2) can also appear depending on the bare coefficients, whereas for d ≥4, only the smooth surface is apparent.

Effects of back reaction in the Brans-Dicke-dilaton cosmology.

The effects of back reaction are studied in the Brans-Dicke-dilaton cosmology in four-dimensional spacetime. In particular, the extended inflationary model is examined by taking account of the effect of back reaction. As quantized matter, conformally invariant quantum fields are considered. In conformally flat spacetime, the expectation value of the stress tensor is exactly calculable. Cosmological solutions are studied in the radiation-dominated and nonvanishing cosmological constant universes. The most important contributions come from the trace-anomaly part of the stress tensor. In most of the parameter region and boundary conditions, solutions behave quite differently from the classical solutions even for the late period of the big bang. Only by fine-tuning the boundary conditions are solutions obtained whose behaviors are close to the classical ones. The sign of the fourth-derivative term of the scale factor in the stress tensor is very important, and it determines whether or not stable solutions exist.

Renormalization of quantum gravity with local GL(4,R) symmetry

Renormalizability of a quantum gravity model with the local GL(4,R) symmetry is investigated. The renormalization conditions based on the action which consists of curvature-squared terms only do not exclude the appearance of fourth-derivative terms in the metric as counterterms. With the addition of all possible, fourth-derivative terms in the metric to the most general curvature-squared Lagrangian, the renormalization becomes complete. The proof is done using the Slavnov-Taylor identities and the power counting argument. Here, a higher-derivative gauge fixing for the general coordinate transformation group is used. Unitarity of the model is not shown.