Higher-order Spatial Derivatives Research Articles

Decay of the compressible Navier-Stokes equations with hyperbolic heat conduction

This paper is concerned with the global well-posedness and large-time behavior of the compressible Navier-Stokes equations with hyperbolic heat conduction, where the heat flux is assumed to satisfy Cattaneo's law. When the initial perturbation is small and regular, we establish the global existence of the classical solution without any additional assumption on the relaxation time τ. Furthermore, assuming the initial perturbation belongs to a negative Sobolev space, we obtain the optimal time-decay rates of the high-order spatial derivatives of solutions by using a pure energy method instead of complicated spectral analysis. It is also observed that the heat flux, due to its damping structure, decays to the motionless state at a faster time-decay rate compared to density, velocity, and temperature.

Vibration behavior and power transmission of coupled plate structures with embedded acoustic black holes joined at an arbitrary angle

The Acoustic Black Hole (ABH) technology is widely recognized for its passive vibration and noise attenuation capabilities, characterized by lightweight and high efficiency. Exploring the dynamic behavior and energy transmission characteristics of the ABH coupled-plate system is a novel endeavor that provides a theoretical foundation for applying ABH structures in complex coupled systems. In this work, a semi-analytical dynamic model for the ABH coupled-plate system (comprising an ABH plate and a uniform plate) with an arbitrary connection angle is proposed by using the Rayleigh-Ritz framework. To ensure the continuity of high-order spatial derivatives at the edges, the modified version of the Fourier series is employed to describe the vibration displacement fields for transverse and in-plane vibration components. This enables the proposed model to determine the necessary derivatives of relevant quantities, namely the structural intensity and power flow, which facilitates the description of energy transmission behaviors in the ABH coupled-plate structure. Firstly, the convergence characteristics of the model are studied and the accuracy of modal information and dynamic response is verified through the comparison of those calculated from the finite element method (FEM), additionally, the experiment is carried out to verify the proposed model. Subsequently, several numerical examples are presented to study the vibrational characteristics of the ABH coupled-plate system and the good vibration attenuation of the ABH coupled-plate is observed as well. Finally, the vibrational energy transmission results are obtained and the influence of some important system parameters such as coupling angles on energy transmission characteristics in such systems are investigated. This work can shed some new light on ABH structures with potential application to such complex coupled-plate structures.

Decay of the 3D Lüst model

In this paper, we consider the time-decay rate of the strong solution to the Cauchy problem for the three-dimensional Lüst model. In particular, the optimal decay rates of the higher-order spatial derivatives of the solution are obtained. The dot{H}^{-s} (0leq s<frac{3}{2}) negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rates.

Dynamic modeling of vibration behavior and power flow of a plate structure embedded with an ABH indentation

A two-dimensional (2D) semi-analytical model is established for the analyses of vibration behavior and energy transmission of the plate structure with a power-law-profiled thickness variation, referred to as an Acoustic Black Hole (ABH) plate. Under the general Rayleigh-Ritz framework, a modified version of 2D Fourier series is employed as the admissible function of the transverse displacement of the ABH plate featuring a thickness variation. Thanks to such modified Fourier series, a high-order spatial derivative continuity over the entire solution domain is available to express the internal forces and bending moments, further calculating the corresponding structural intensity and power flow within the ABH plate. The proposed model allows an accurate prediction of the modal characteristics and dynamic response, which is verified by the FEM and experiment, respectively. Several numerical analyses are proceeded and the typical dynamic phenomena, including the wave compression and efficient vibrational suppression of the 2D ABH plate are observed. The structural intensity (SI) vector arrows of the ABH plate are given, which represent the magnitude of the vibrational energy and the flowing direction in a visual pattern. The energy transmission behaviors and the manipulation of power flow by the ABH are revealed in an energy perspective. The typical energy focalization and dissipation phenomena of the ABH plate are analyzed in the frequency domain. Finally, the enhancement of the energy transmission by the ABH feature on the plate is explored by the normalized power flow. It is noted that the proposed model offers a visual energy transfer analysis tool for the 2D ABH structures, which may provide a novel perspective in various studies of ABH structures.

Well-posed evolution of field theories with anisotropic scaling: the Lifshitz scalar field in a black hole space-time

Partial differential equations exhibiting an anisotropic scaling between space and time — such as those of Hořava-Lifshitz gravity — have a dispersive nature. They contain higher-order spatial derivatives, but remain second order in time. This is inconvenient for performing long-time numerical evolutions, as standard explicit schemes fail to maintain convergence unless the time step is chosen to be very small. In this work, we develop an implicit evolution scheme that does not suffer from this drawback, and which is stable and second-order accurate. As a proof of concept, we study the numerical evolution of a Lifshitz scalar field on top of a spherically symmetric black hole space-time. We explore the evolution of a static pulse and an (approximately) ingoing wave-packet for different strengths of the Lorentz-breaking terms, accounting also for the effect of the angular momentum eigenvalue and the resulting effective centrifugal barrier. Our results indicate that the dispersive terms produce a cascade of modes that accumulate in the region in between the Killing and universal horizons, indicating a possible instability of the latter.

A novel denoising strategy for ultrasound elastography

Ultrasound imaging, while offering real-time tissue monitoring, portability, and other benefits, often suffers from poor signal to noise ratio (SNR). Noise has a particularly detrimental effect in US elastography since it gets amplified during the elastic modulus reconstruction process due to computation of high-order spatial derivatives of the deformation field needed to calculate the shear wave velocity within the sample. In this work, we introduce a novel scheme to denoise deformation fields using physically motivated “compatibility conditions,” which are imposed as constraints. These conditions, well established in the solid mechanics literature, ensure that the deformation field remains continuous without exhibiting cracks, folds, or slips. Spatially uncorrelated noise gets significantly reduced when we apply them. Importantly, this approach avoids unnecessary smoothing and the ensuing loss of accuracy commonly observed in other denoising methods. We demonstrate the efficacy of this method using planar deformation fields obtained from simulations and those measured on an elasticity quality assurance (QA) phantom using acoustic radiation force impulse imaging (ARFI). Our pipeline results in smoother elasticity maps with 2x improved SNR, confirming the effectiveness of the approach.

Some Traveling Wave Solutions to the Fifth-Order Nonlinear Wave Equation Using Three Techniques: Bernoulli Sub-ODE, Modified Auxiliary Equation, and G ′ / G -Expansion Methods

The fifth-order nonlinear wave equation contains terms involving higher-order spatial derivatives, such as u x x x and u x x x x x . These terms are responsible for dispersion, which affects the shape and propagation of the wave. The study of dispersion is important in many areas, including seismology, acoustics, and communication theory. In the current work, three potent analytical techniques are proposed in order to solve the fifth-order nonlinear wave equation. The used approaches are the modified auxiliary equation method, the Bernoulli Sub-ODE method, and the G ′ / G -expansion method (MAE). Some graphs are plotted to display our findings. The solutions to the nonlinear wave equation are used to describe the nonlinear dynamics of waves in physical systems. The results show how the dynamics of the wave solutions are influenced by the system parameters, which can be used as system controllers. The new approaches used in this work helped to find new solutions for traveling waves. This could be seen as a new contribution to the field. Water waves, plasma waves, and acoustic wave behavior can be described by the obtained solutions.

Joint Coefficient and Solution Estimation for the 1-D Wave Equation: An Observer-Based Solution to Inverse Problems

The estimation of an unknown source coefficient and of the solution to a 1-dimensional (1D) wave equation via exponentially convergent state observers is the problem under consideration in this work. The coefficient is assumed to depend on the space variable only and to be polynomial. The main observation information for this inverse problem is the value of the solution to the wave equation in a subinterval of the domain, including also some of its higher-order spatial derivatives. In order to estimate the source coefficient, we turn it into a new state as in finite-dimensional parameter identification approaches. However in this infinite-dimensional setting, this requires the introduction of a novel indirect approach involving an infinite-dimensional state transformation. Sufficient conditions allow the design of a composite observer consisting of an internal observer, which estimates in higher regularity spatial norms both the source term and the solution on a subinterval, and a boundary observer, in order to eventually provide the estimation of the solution everywhere. The observer convergence is proven by means of Lyapunov analysis. An extension of this approach to the case of the identification of a potential in the wave equation is finally considered, which is a nonlinear inverse problem since here, in the equation, the unknown coefficient multiplies the solution.

“Implicit” vs “Explicit” gradient plasticity models: Do they always remove mesh dependence in softening materials?

Finite element quasi-static solutions of rate-independent softening plasticity models are known to depend on the mesh size (i.e., are unreliable) due to loss of ellipticity of the governing partial differential equations, which allows for the development of non-smooth solutions, such as shear bands of zero thickness. A “regularization” scheme that has been proposed is the use of “non-local” (Bažant et al., 1984) or “strain-gradient” theories (Aifantis, 1984). We examine in detail the families of “explicit” and “implicit” gradient plasticity models and show that use of such theories does not always eliminate mesh dependence of finite element solutions in rate-independent softening materials. Depending on the value of the non-local hardening modulus, the mathematical problem may still lose ellipticity and allow for the development of solutions with discontinuous spatial velocity gradients, which are responsible for the mesh dependence of numerical solutions. As an example, a rate-independent “implicit” gradient version of the von Mises yield condition with the associated flow rule is implemented in ABAQUS, and the formation of shear bands in plane strain tension is analyzed numerically; it is shown that, unless the material parameters in the non-local model are such that ellipticity is retained, the finite element solutions depend on the mesh size. The example of isotropic porous metal plasticity with a hardening matrix material is considered (e.g., the Gurson (1977) model); it is shown that regularization is achieved if an implicit non-local model is used, in which the local porosity f is replaced by its non-local (neighborhood average) counterpart f̄ in the yield condition. In all non-local plasticity models, dimensional consistency requires the introduction of one or more “material lengths” ℓi, which multiply the additional terms with the higher-order spatial derivatives introduced by the non-local models. The boundary layers associated with the singular perturbation problem when the material lengths ℓi→0 are examined. A linear stability analysis is carried out to study the stability of homogeneous problems under small harmonic perturbations; it is shown that the stability range of the non-local models is always larger than that of the corresponding local models.

Global existence and time-decay rates of classical solutions to the generalized incompressible Oldroyd-B model in [formula omitted]

In this paper, we investigate the initial value problem for the three dimensional generalized incompressible Oldroyd-B model. Based on the energy estimates and the tricky analytical skills, we prove that the problem always exists a unique global classical solution without any smallness restriction on the higher order spatial derivatives of initial data. Moreover, the optimal time-decay rates of solutions itself and the higher-order spatial derivatives of solutions are obtained without linear decay analysis. The result implies that the negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rate.

Numerical Study on Viscous Fingering Using Electric Fields in a Hele-Shaw Cell

We investigate the nonlinear dynamics of a moving interface in a Hele-Shaw cell subject to an in-plane applied electric field. We develop a spectrally accurate boundary integral method where a coupled integral equation system is formulated. Although the stiffness due to the high order spatial derivatives can be removed, the long-time simulation is still expensive since the evolving velocity of the interface drops dramatically as the interface expands. We remove this physically imposed stiffness by employing a rescaling scheme, which accelerates the slow dynamics and reduces the computational cost. Our nonlinear results reveal that positive currents restrain finger ramification and promote overall stabilization of patterns. On the other hand, negative currents make the interface more unstable and lead to the formation of thin tail structures connecting the fingers and a small inner region. When no flux is injected, and a negative current is utilized, the interface tends to approach the origin and break up into several drops. We investigate the temporal evolution of the smallest distance between the interface and the origin and find that it obeys an algebraic law $\displaystyle (t_*-t)^b$, where $t_*$ is the estimated pinch-off time.

High-Order Semi-implicit Schemes for Evolutionary Partial Differential Equations with Higher Order Derivatives

The aim of this work is to apply a semi-implicit (SI) strategy in an implicit-explicit (IMEX) Runge–Kutta (RK) setting introduced in Boscarino et al. (J Sci Comput 68:975–1001, 2016) to a sequence of 1D time-dependent partial differential equations (PDEs) with high order spatial derivatives. This strategy gives a great flexibility to treat these equations, and allows the construction of simple linearly implicit schemes without any Newton’s iteration. Furthermore, the SI IMEX-RK schemes so designed does not need any severe time step restriction that usually one has using explicit methods for the stability, i.e. Delta t = {mathcal {O}}(Delta t^k) for the kth (k ge 2) order PDEs. For the space discretization, this strategy is combined with finite difference schemes. We illustrate the effectiveness of the schemes with many applications to dissipative, dispersive and biharmonic-type equations. Numerical experiments show that the proposed schemes are stable and can achieve optimal orders of accuracy.

Space-time decay rate of high-order spatial derivative of solution for 3D compressible Euler equations with damping

&lt;abstract&gt;&lt;p&gt;We are concerned with the space-time decay rate of high-order spatial derivatives of solutions for 3D compressible Euler equations with damping. For any integer $ \ell\geq3 $, Kim (2022) showed the space-time decay rate of the $ k(0\leq k\leq \ell-2) $th-order spatial derivative of the solution. By making full use of the structure of the system, and employing different weighted energy methods for $ 0\leq k \leq \ell-2, k = \ell-1, k = \ell $, it is shown that the space-time decay rate of the $ (\ell-1) $th-order and $ \ell $th-order spatial derivative of the strong solution in weighted Lebesgue space $ L_\sigma^2 $ are $ t^{-\frac{3}{4}-\frac{\ell-1}{2}+\frac{\sigma}{2}} $ and $ t^{-\frac{3}{4}-\frac{\ell}{2}+\frac{\sigma}{2}} $ respectively, which are totally new as compared to that of Kim (2022) &lt;sup&gt;[&lt;xref ref-type="bibr" rid="b1"&gt;1&lt;/xref&gt;]&lt;/sup&gt;.&lt;/p&gt;&lt;/abstract&gt;

Finite line method for solving high-order partial differential equations in science and engineering

In this paper, a completely new numerical method, named Finite Line Method (FLM), is proposed for solving general linear and non-linear high-order partial differential equations (PDEs) in science as well as engineering problems in heat conduction and mechanics. In this method, the computational domain is discretized into a number of collocation points as in the free element method (Gao et al., 2019; Liu et al., 2019), and at each collocation point a set of straight or curved lines crossing the point is formed, which is represented by a number of nodes distributed over lines. The shape functions for each line are constructed using the Lagrange interpolation formulation and the first and high order spatial partial derivatives of the shape functions with respect to global coordinates are derived through an ingenious technique. The derived partial derivatives can be directly substituted into the governing differential equations and related boundary conditions to set up the discretized system of equations.FLM is a type of collocation method, not needing any integration to establish the solution scheme. And since the Lagrange interpolation formulation is used to construct the shape functions, arbitrarily high order lines can be easily formulated for solving any types of linear and non-linear high order PDEs and engineering problems. A number of numerical examples for PDEs, heat conduction and solid mechanics problems are given to demonstrate the correctness and stability of the proposed method.

Global well-posedness and large time behavior of epitaxy thin film growth model

We consider the global well-posedness and large time behavior of solutions for epitaxy thin film growth model in mathbb{R}^{d} with the dimensional dgeq 3. First, using the pure energy method and a standard continuity argument, we prove that there exists a unique global strong solution under the condition that the initial data is sufficiently small. Moreover, we also establish the suitable negative Sobolev norm estimates and obtain the optimal decay rates of the higher-order spatial derivatives of the strong solution.

Optimal large time behaviour of the 3D compressible magnetohydrodynamics equations with large initial data

We investigate time-decay rates of strong solutions to the 3D compressible magnetohydrodynamics equations with large initial data. The main novelty of this paper is two-fold: first, we prove the upper optimal decay rates of the higher-order spatial derivatives of the solution, which are the same as those of the heat equation, and faster than the decay rates in the previous related works. Second, if the initial data satisfy some additional low frequency assumption, we also show the lower optimal decay rates of the solution as well as its all-order spatial derivatives. Therefore, our decay rates are optimal in this sense. Our methods mainly involve the Fourier splitting method, low-frequency and high-frequency decomposition and delicate energy estimates.

Low-order continualization of an anisotropic membrane lattice with next-nearest interactions. Enhanced prediction of its dynamic behaviour

In this paper, a novel anisotropic membrane lattice with nearest and next-nearest interactions (long-range forces) has been continualized through different standard and non-standard continualization procedures, which enables the development of new non-classical continuum models capable of accurately capturing the scale effects, present in the matter due to its discrete nature. The performance of these continuum models is assessed by means of both dispersion and natural frequencies analyses, where the discrete model is considered as a reference. In addition, the appearance of certain physical inconsistencies in some of the developed models is analysed, concluding that these only appear for those developed with continualizations based on Taylor expansion. Interestingly, the non-standard models suitably capture the dispersive behaviour of the discrete one, without both physical inconsistencies and higher-order spatial derivatives, thus avoiding the need for extra boundary conditions when finite solids are involved.

Accuracy-Enhancement of Discontinuous Galerkin Methods for PDEs Containing High Order Spatial Derivatives

Accuracy-Enhancement of Discontinuous Galerkin Methods for PDEs Containing High Order Spatial Derivatives

Decay rates for the higher-order derivatives of the Navier-Stokes flow in exterior domains

This paper makes investigations on the time-decay rates for the 3-dimensional Navier-Stokes flow in exterior domains. Boundedness of the convolution operator with the Gaussian kernel from the intersection L01(Ω)∩L1(Ω,|x|αdx) for some 0<α≤1 to the space Lr(Ω) for all 1≤r<∞ is firstly established, from which, an innovated estimate ‖u(t)‖r≤Ct−α2−32(1−1r) (t≥T0, 1<r≤∞) is derived. Decay rates of the higher-order spatial and temporal derivatives of the flow are mainly studied then. In order to derive the estimates ‖∇∂tku(t)‖r≤Ct−k+12−α2−32(1−1r) adequately, translation transformation v˜kv(t)=∂tku(t+Tk) has been made.

High-order strongly nonlinear long wave approximation and solitary wave solution

We consider high-order strongly nonlinear long wave models expanded in a single small parameter measuring the ratio of the water depth to the characteristic wavelength. By examining its dispersion relation, the high-order system for the bottom velocity is found stable to all disturbances at any order of approximation. On the other hand, systems for other velocities can be unstable and even ill-posed, as signified by the unbounded maximum growth. Under the steady assumption, a new third-order solitary wave solution of the Euler equations is obtained using the high-order strongly nonlinear system and is expanded in an amplitude parameter, which is different from that used in weakly nonlinear theory. The third-order solution is shown to well describe various physical quantities induced by a finite-amplitude solitary wave, including the wave profile, horizontal velocity profile, particle velocity at the crest and bottom pressure. For numerical computations, the first- and second-order strongly nonlinear systems for the bottom velocity are considered. It is shown that finite difference schemes are unstable due to truncation errors introduced in approximating high-order spatial derivatives and, therefore, a more accurate spatial discretization scheme is necessary. Using a pseudo-spectral method based on finite Fourier series combined with an iterative scheme for the inversion of a non-local operator, the strongly nonlinear systems are solved numerically for the propagation of a single solitary wave and the head-on collision of two counter-propagating solitary waves of finite amplitudes, and the results are compared with previous laboratory measurements.