Second-order Spatial Derivatives Research Articles

Regularization for a Riesz-Feller space fractional backward diffusion problem with a time-dependent coefficient

In the present paper, we consider a backward problem for a space-fractional diffusion equation (SFDE) with a time-dependent coefficient. Such the problem is obtained from the classical diffusion equation by replacing the second-order spatial derivative with the Riesz-Feller derivative of order α∈(0,2]. This problem is ill-posed, i.e., the solution (if it exists) does not depend continuously on the data. Therefore, we propose one new regularization solution to solve it. Then, the convergence estimate is obtained under a priori bound assumptions for exact solution.

Nonsplit complex-frequency shifted perfectly matched layer combined with symplectic methods for solving second-order seismic wave equations — Part 1: Method

The absorbing boundary condition plays an important role in seismic wave modeling. The perfectly matched layer (PML) boundary condition has been established as one of the most effective and prevalent absorbing boundary conditions. Among the existing PML-type conditions, the complex frequency shift (CFS) PML attracts considerable attention because it can handle the evanescent and grazing waves better. For solving the resultant CFS-PML equation in the time domain, one effective technique is to apply convolution operations, which forms the so-called convolutional PML (CPML). We have developed the corresponding CPML conditions with nonconstant grid compression parameter, and used its combination algorithms specifically with the symplectic partitioned Runge-Kutta and the nearly analytic SPRK methods for solving second-order seismic wave equations. This involves evaluating second-order spatial derivatives with respect to the complex stretching coordinates at the noninteger time layer. Meanwhile, two kinds of simplification algorithms are proposed to compute the composite convolutions terms contained therein.

Numerical analysis and pattern formation process for space-fractional superdiffusive systems

In this paper, we consider the numerical solution of fractional-in-space reaction-diffusion equation, which is obtained from the classical reaction-diffusion equation by replacing the second-order spatial derivative with a fractional derivative of order \begin{document}$ α∈(1, 2] $\end{document} . We adopt a class of second-order approximations, based on the weighted and shifted Grunwald difference operators in Riemann-Liouville sense to numerically simulate two multicomponent systems with fractional-order in higher dimensions. The efficiency and accuracy of the numerical schemes are justified by reporting the norm infinity and norm relative errors as well as their convergence. The complexity of the dynamics in the equation is theoretically discussed by conducting its local and global stability analysis and Numerical experiments are performed to back-up the theoretical claims.

An efficient ray-tracing method and its application to Gaussian beam migration in complex multilayered anisotropic media

We have developed a fast ray-tracing method for multiple layered inhomogeneous anisotropic media, based on the generalized Snell’s law. Realistic geologic structures continuously varying with embedded discontinuities are parameterized by adopting cubic B-splines with nonuniformly spaced nodes. Because the anisotropic characteristic is often closely related to the interface configuration, this model parameterization scheme containing the natural inclination of the corresponding layer is particularly suitable for tilted transverse isotropic models whose symmetry axis is generally perpendicular to the direction of the layers. With this model parameterization, the first- and second-order spatial derivatives of the velocity within the interfaces can be effectively obtained, which facilitates the amplitude computation in dynamic ray tracing. By using complex initial conditions for the dynamic ray system and taking the multipath effect into consideration, our method is applicable to Gaussian beam migration. Numerical experiments of our method have been used to verify its effectiveness, practicability, and efficiency in memory storage and computation.

Poincaré inequalities for Sobolev spaces with matrix-valued weights and applications to degenerate partial differential equations

For bounded domains Ω, we prove that the Lp-norm of a regular function with compact support is controlled by weighted Lp-norms of its gradient, where the weight belongs to a class of symmetric non-negative definite matrix-valued functions. The class of weights is defined by regularity assumptions and structural conditions on the degeneracy set, where the determinant vanishes. In particular, the weight A is assumed to have rank at least 1 when restricted to the normal bundle of the degeneracy set S. This generalization of the classical Poincaré inequality is then applied to develop a robust theory of first-order Lp-based Sobolev spaces with matrix-valued weight A. The Poincaré inequality and these Sobolev spaces are then applied to produce various results on existence, uniqueness and qualitative properties of weak solutions to boundary-value problems for degenerate elliptic, degenerate parabolic and degenerate hyperbolic partial differential equations (PDEs) of second order written in divergence form, where A is calibrated to the matrix of coefficients of the second-order spatial derivatives. The notion of weak solution is variational: the spatial states belong to the matrix-weighted Sobolev spaces with p = 2. For the degenerate elliptic PDEs, the Dirichlet problem is treated by the use of the Poincaré inequality and Lax–Milgram theorem, while the treatment of the Cauchy–Dirichlet problem for the degenerate evolution equations relies only on the Poincaré inequality and the parabolic and hyperbolic counterparts of the Lax–Milgram theorem.

A new central compact finite difference scheme with high spectral resolution for acoustic wave equation

Based on the existing cell-node and cell-centered compact finite difference schemes, we developed a new central compact scheme with a high spectral resolution for the acoustic wave equation. In the new scheme, both the function values on the cell-nodes and cell-centers are used to compute the second-order spatial derivatives on the cell-nodes. The cell-centered values are stored and updated as independent variables in the modeling. The spatial derivatives on the cell-centers are evaluated by half shifting the indices in the formula designed for the cell-nodes. Compared to the conventional compact interpolation scheme, the proposed approach can avoid introducing transfer errors. Either Taylor-series expansion-based or optimized least-squares-based methods are used to calculate the finite difference coefficients. Theoretical analysis and synthetic examples demonstrate that the optimized least-squares-based method can provide higher accuracy than the Taylor-series expansion-based method. This new scheme is not a simple combination of the cell-node and cell-centered compact schemes and outperforms them in three scenarios. Firstly, it can promise higher accuracy considering the same formal truncation errors and model parameters. Thus, it can maintain superior precision while using a shorter spatial finite difference stencil. Secondly, compared to the cell-node compact scheme with half of grid spacing, the new scheme can yield equally as accurate results with less time consuming, together with saving approximately 25% and 29% of memory in 2D and 3D modeling, respectively. Finally, for similar memory requirements, the new method can more efficiently provide solutions with higher accuracy. The synthetic examples on the 2D homogeneous and the 3D horizontally-layered models demonstrate the advantages of the proposed scheme. The numerical simulations with 2D Marmousi model further validate its accuracy, efficiency and flexibility in complex media.

Plasmonic circuit for second-order spatial differentiation at the subwavelength scale.

We suggest a plasmonic nanodevice for performing the second-order spatial derivative of light fields. The device consists of five gold nanorods arranged to evanescently couple to each other so that emit cross-polarized output proportional to the second-order differentiation of the incident wave. A theoretical model based on the electrostatic eigenmode analysis is derived and numerical simulations using the finite-difference time-domain methods are provided as supporting evidence. It is shown in both the analytic and numerical methods that the proposed plasmonic circuit performs second-order differentiation of the phase of the incident light field in transmission mode with a subwavelength planar resolution. The resolution of 0.29 λ-1 is numerically demonstrated for a 20 nm thick circuit at the wavelength of 700 nm. The suggested plasmonic device has potential application in miniaturized systems for all-optical computation.

The Bassi Rebay 1 scheme is a special case of the Symmetric Interior Penalty formulation for discontinuous Galerkin discretisations with Gauss–Lobatto points

In the discontinuous Galerkin (DG) community, several formulations have been proposed to solve PDEs involving second-order spatial derivatives (e.g. elliptic problems). In this paper, we show that, when the discretisation is restricted to the usage of Gauss–Lobatto points, there are important similarities between two common choices: the Bassi-Rebay 1 (BR1) method, and the Symmetric Interior Penalty (SIP) formulation. This equivalence enables the extrapolation of properties from one scheme to the other: a sharper estimation of the minimum penalty parameter for the SIP stability (compared to the more general estimate proposed by Shahbazi [1]), more efficient implementations of the BR1 scheme, and the compactness of the BR1 method for straight quadrilateral and hexahedral meshes.

A nonlocal species concentration theory for diffusion and phase changes in electrode particles of lithium ion batteries

A nonlocal species concentration theory for diffusion and phase changes is introduced from a nonlocal free energy density. It can be applied, say, to electrode materials of lithium ion batteries. This theory incorporates two second-order partial differential equations involving second-order spatial derivatives of species concentration and an additional variable called nonlocal species concentration. Nonlocal species concentration theory can be interpreted as an extension of the Cahn–Hilliard theory. In principle, nonlocal effects beyond an infinitesimal neighborhood are taken into account. In this theory, the nonlocal free energy density is split into the penalty energy density and the variance energy density. The thickness of the interface between two phases in phase segregated states of a material is controlled by a normalized penalty energy coefficient and a characteristic interface length scale. We implemented the theory in COMSOL Multiphysics\(^{\circledR }\) for a spherically symmetric boundary value problem of lithium insertion into a \(\hbox {Li}_x\hbox {Mn}_2\hbox {O}_4\) cathode material particle of a lithium ion battery. The two above-mentioned material parameters controlling the interface are determined for \(\hbox {Li}_x\hbox {Mn}_2\hbox {O}_4\), and the interface evolution is studied. Comparison to the Cahn–Hilliard theory shows that nonlocal species concentration theory is superior when simulating problems where the dimensions of the microstructure such as phase boundaries are of the same order of magnitude as the problem size. This is typically the case in nanosized particles of phase-separating electrode materials. For example, the nonlocality of nonlocal species concentration theory turns out to make the interface of the local concentration field thinner than in Cahn–Hilliard theory.

Positivity-preserving sixth-order implicit finite difference weighted essentially non-oscillatory scheme for the nonlinear heat equation

This paper presents a class of semi-implicit finite difference weighted essentially non-oscillatory (WENO) schemes for solving the nonlinear heat equation. For the discretization of second-order spatial derivatives, a sixth-order modified WENO scheme is directly implemented. This scheme preserves the positivity principle and rejects spurious oscillations close to non-smooth points. In order to admit large time steps, a class of implicit Runge–Kutta methods is used for the temporal discretization. The implicit parts of these methods are linearized in time by using the local Taylor expansion of the flux. The stability analysis of the semi-implicit WENO scheme with 3-stages form is provided. Finally, some comparative results for one-, two- and three-dimensional PDEs are included to illustrate the effectiveness of the proposed approach.

Linear distributed source modeling of local field potentials recorded with intra-cortical electrode arrays.

Planar intra-cortical electrode (Utah) arrays provide a unique window into the spatial organization of cortical activity. Reconstruction of the current source density (CSD) underlying such recordings, however, requires “inverting” Poisson’s equation. For inter-laminar recordings, this is commonly done by the CSD method, which consists in taking the second-order spatial derivative of the recorded local field potentials (LFPs). Although the CSD method has been tremendously successful in mapping the current generators underlying inter-laminar LFPs, its application to planar recordings is more challenging. While for inter-laminar recordings the CSD method seems reasonably robust against violations of its assumptions, is it unclear as to what extent this holds for planar recordings. One of the objectives of this study is to characterize the conditions under which the CSD method can be successfully applied to Utah array data. Using forward modeling, we find that for spatially coherent CSDs, the CSD method yields inaccurate reconstructions due to volume-conducted contamination from currents in deeper cortical layers. An alternative approach is to “invert” a constructed forward model. The advantage of this approach is that any a priori knowledge about the geometrical and electrical properties of the tissue can be taken into account. Although several inverse methods have been proposed for LFP data, the applicability of existing electroencephalographic (EEG) and magnetoencephalographic (MEG) inverse methods to LFP data is largely unexplored. Another objective of our study therefore, is to assess the applicability of the most commonly used EEG/MEG inverse methods to Utah array data. Our main conclusion is that these inverse methods provide more accurate CSD reconstructions than the CSD method. We illustrate the inverse methods using event-related potentials recorded from primary visual cortex of a macaque monkey during a motion discrimination task.

A conservative, spatially continuous method of lines for one-dimensional reaction-diffusion equations

PurposeThe purpose of this paper is to develop a new finite-volume method of lines for one-dimensional reaction-diffusion equations that provides piece-wise analytical solutions in space and is conservative, compare it with other finite-difference discretizations and assess the effects of the nonlinear diffusion coefficient on wave propagation.Design/methodology/approachA conservative, finite-volume method of lines based on piecewise integration of the diffusion operator that provides a globally continuous approximate solution and is second-order accurate is presented. Numerical experiments that assess the accuracy of the method and the time required to achieve steady state, and the effects of the nonlinear diffusion coefficients on wave propagation and boundary values are reported.FindingsThe finite-volume method of lines presented here involves the nodal values and their first-order time derivatives at three adjacent grid points, is linearly stable for a first-order accurate Euler’s backward discretization of the time derivative and has a smaller amplification factor than a second-order accurate three-point centered discretization of the second-order spatial derivative. For a system of two nonlinearly-coupled, one-dimensional reaction-diffusion equations, the amplitude, speed and separation of wave fronts are found to be strong functions of the dependence of the nonlinear diffusion coefficients on the concentration and temperature.Originality/valueA new finite-volume method of lines for one-dimensional reaction-diffusion equations based on piecewise analytical integration of the diffusion operator and the continuity of the dependent variables and their fluxes at the cell boundaries is presented. The method may be used to study heat and mass transfer in layered media.

High-precision acoustic modeling with second-order staggered difference

Since the precision of spatial derivative of wavefield calculated by staggered difference is superior to that by central difference, the formula of second-order staggered differential coefficient with 2N-order precision is constructed via mathematical deduction on combination of first-order differential coefficient on each grid point, and its validity is verified by numerical simulation in constant velocity media. With the staggered mesh, the maximum peak frequency of Ricker wavelet which does not generate numerical dispersion may be increased by 10–15 Hz, thus the vertical resolution of simulated seismograms may be enhanced. In order to eliminate the boundary reflection which is characterized as high frequency, the absorbing boundary condition of perfectly matched layer with staggered mesh is utilized in numerical simulation of scalar-wave equation, and the suppression effect is superior to that of the central difference. The precision of simulated records may be improved when staggered mesh is adopted in calculation of the second-order spatial derivative, and the ringing phenomena appeared before the first break may be avoided. The computational cost of second-order staggered difference increased by only 25% on 3.60 GHz Intel Core i7–4790 processor when the precision of simulated records is same as that simulated by second-order central difference.

The error in solving a wave equation based on a scheme with weights

The error in approximating a Cauchy problem for a two-dimensional wave equation based on a scheme with weights is studied. The dependence of the approximation error on the time step and weight parameter is considered. With this aim in view, the difference of second-order spatial derivatives in the wave equation is approximated, while the time derivative is preserved continuous; and an analytical solution of a Cauchy problem for a system of ordinary differential equations is obtained as the decomposition in the orthonormal basis consisting of the eigenvectors of the operator of the second difference derivative with respect to the spatial variables. Based on this solution, the errors of the approximation of the wave problem by three-layer difference schemes are studied and the conditions for the stability of a three-layer difference scheme are obtained. It is established that, when simulating the propagation of oscillation processes using difference methods, the oscillation frequency values differ from the real ones and depend on the weight parameter and time step. The optimal values of the weight parameter with which the deviation of the oscillation frequency for the difference scheme is minimal are obtained. The dependences of the approximation error on weight and spatial step are derived. The optimal values of the weight parameter with which the schemes are of the second and fourth order of accuracy with respect to the time step are found.

On the thermodynamics of the Swift–Hohenberg theory

We present the microbalance including the microforces, the first- and second-order microstresses for the Swift--Hohenberg equation concomitantly with their constitutive equations, which are consistent with the free-energy imbalance. We provide an explicit form for the microstress structure for a free-energy functional endowed with second-order spatial derivatives. Additionally, we generalize the Swift--Hohenberg theory via a proper constitutive process. Finally, we present one highly-resolved three-dimensional numerical simulation to demonstrate the particular form of the resulting microstresses and their interactions in the evolution of the Swift--Hohenberg equation.

Time-linearized, compact methods for the inviscid GRLW equation subject to initial Gaussian conditions

• Effects of the width of initial Gaussian conditions on wave propagation are studied. • Roles of nonlinearity and local advection on wave dynamics are investigated. • Time-linearization methods for the GRLW equation are presented. • Rarefaction waves may appear behind solitary waves. • Localized dispersive oscillations may occur behind the leading wave. The objective of this paper is three-fold. First, four time-linearization methods that are second- and fourth-order accurate in time and space, respectively, are presented and used to study the dynamics of the modified and generalized regularized-long wave equations (mRLW and GRLW equations, respectively). Two of the methods use the conservation-law form of the equations and treat the wave amplitude and its second-order spatial derivative and the linear and nonlinear advection fluxes as unknowns, whereas the other two employ the non-conservation-law form of the equations and consider the wave amplitude and its first- and second-order spatial derivatives as unknowns. The methods employ three-point fourth-order accurate Padé discretizations for the first- and second-order derivatives, are second-order accurate in time, and yield linear systems of blocktridiagonal matrices. Second, the accuracy of these methods is assessed by comparing their results with those of the exact solution of the mRLW equation. It is reported that the four methods predict nearly the same values of the three invariants and have the same accuracy, and that an accurate prediction of the invariants may not correspond to small errors in space and time. Third, the dynamics of the inviscid GRLW equation is studied first qualitatively in terms of length and time scales and then numerically as a function of the linear advection speed, the exponent of the nonlinear advection flux, the dispersion coefficient and the amplitude and width of the initial bell-shaped or Gaussian conditions. It is shown that wide initial conditions result in wave steepening and breakup and the formation of solitary waves whose amplitude and speed decrease as the time for their formation increases. For narrow initial conditions, it is shown that only a single solitary wave may form. Behind this wave and depending on the parameters that characterized the inviscid GRLW equation, rarefaction or negative amplitude waves that propagate towards the upstream boundary or a train of localized oscillatory waves that do not emerge from the trailing edge of the leading solitary wave may be formed. These oscillatory waves exhibit the characteristics of, but are not dispersive shock waves and their amplitude and frequency increases as the width of the initial conditions is decreased. The results presented here do not only complement previous work by the authors, they also show that the dynamics of the inviscid GRLW equation undergoes new and interesting phenomena as the width of the initial conditions is decreased.

Spatial Model Predictive Control for Smooth and Accurate Steering of an Autonomous Truck

In this paper, we present an algorithm for lateral control of a vehicle—a smooth and accurate model predictive controller (MPC). The fundamental difference compared to a standard MPC is that the driving smoothness is directly addressed in the cost function. The controller objective is based on the minimization of the first- and second-order spatial derivatives of the curvature. By doing so, jerky commands to the steering wheel, which could lead to permanent damage on the steering components and vehicle structure, are avoided. A good path tracking accuracy is ensured by adding constraints to avoid deviations from the reference path. Finally, the controller is experimentally tested and evaluated on a Scania construction truck. The evaluation is performed at Scania's facilities near Sodertalje, Sweden via two different paths: a precision track that resembles a mining scenario and a high-speed test track that resembles a highway situation. Even using a linearized kinematic vehicle to predict the vehicle motion, the performance of the proposed controller is encouraging, since the deviation from the path never exceeds 30 cm. It clearly outperforms an industrial pure-pursuit controller in terms of path accuracy and a standard MPC in terms of driving smoothness.

Acoustic VTI Modeling Using an Optimal Time-Space Domain Finite-Difference Scheme

Finite-difference (FD) schemes have been used widely for solving wave equations in seismic exploration. However, the conventional FD schemes hardly guarantee high accuracy at both small and large wavenumbers. In this paper, we propose an optimal time-space domain FD scheme for acoustic vertical transversely isotropic (VTI) wave modeling. The optimal FD coefficients for the second-order spatial derivatives are derived by approaching the time-space domain dispersion relation of acoustic VTI wave equations with the combination of the Taylor-series expansion and the sampling interpolation. We perform numerical dispersion analyses and acoustic VTI modeling using the optimal time-space domain FD scheme. The numerical dispersion analyses show that the optimal FD scheme has smaller dispersion than the conventional FD scheme at large wavenumbers, and also preserves high accuracy at small wavenumbers. The acoustic VTI modeling examples also demonstrate that the optimal time-space domain FD scheme has greater accuracy compared with the conventional time-space domain FD scheme for the same modeling parameters.

Robust and adaptive techniques for numerical simulation of nonlinear partial differential equations of fractional order

In this paper, some nonlinear space-fractional order reaction-diffusion equations (SFORDE) on a finite but large spatial domain x ∈ [0, L], x=x(x,y,z) and t ∈ [0, T] are considered. Also in this work, the standard reaction-diffusion system with boundary conditions is generalized by replacing the second-order spatial derivatives with Riemann-Liouville space-fractional derivatives of order α, for 0 < α < 2. Fourier spectral method is introduced as a better alternative to existing low order schemes for the integration of fractional in space reaction-diffusion problems in conjunction with an adaptive exponential time differencing method, and solve a range of one-, two- and three-components SFORDE numerically to obtain patterns in one- and two-dimensions with a straight forward extension to three spatial dimensions in a sub-diffusive (0 < α < 1) and super-diffusive (1 < α < 2) scenarios. It is observed that computer simulations of SFORDE give enough evidence that pattern formation in fractional medium at certain parameter value is practically the same as in the standard reaction-diffusion case. With application to models in biology and physics, different spatiotemporal dynamics are observed and displayed.

Analysis of the compact difference scheme for the semilinear fractional partial differential equation with time delay

In the paper, a linearized compact finite difference scheme is presented for the semilinear fractional delay convection-reaction–diffusion equation. Firstly, the equation is transformed into an equivalent semilinear fractional delay reaction–diffusion equation by using a special transformation. Then, the temporal Caputo derivative is discreted by using approximation and the second-order spatial derivative is approximated by the compact finite difference scheme. The solvability, unconditional stability, and convergence in the sense of - and - norms are proved rigorously. Finally, numerical examples are carried out extensively to support our theoretical analysis.