Pseudospectral Scheme Research Articles

Analysis of a Fourier pseudo-spectral conservative scheme for the Klein–Gordon–Schrödinger equation

ABSTRACTIn this paper, we focus on constructing and analysing a new Fourier pseudo-spectral conservative scheme for the Klein–Gordon–Schrödinger (KGS) equation. After rewriting the KGS equation as an infinite-dimensional Hamiltonian system, we use a Fourier pseudo-spectral method to discrete the system in space to obtain a semi-discrete system, which can be cast into a canonical finite-dimensional Hamiltonian form. Then, an energy-preserving and charge-preserving scheme is constructed by using the symmetric discrete gradient method. Based on the discrete conservation laws and the equivalence of the semi-norm between the Fourier pseudo-spectral method and the finite difference method, the pseudo-spectral solution of the proposed scheme is proved to be bounded in the discrete norm. The proposed scheme is shown to be convergent with the convergence order of in the discrete norm afterwards, where J is the number of nodes and τ is the time step size. Numerical experiments are conducted to verify the theoretical analysis.

Critical behavior of the black hole/black string transition

We numerically construct static localized black holes in five and six spacetime dimensions which are solutions to Einstein’s vacuum field equations with one compact periodic dimension. In particular, we investigate the critical regime in which the poles of the localized black hole are about to merge. A well adapted multi-domain pseudo-spectral scheme provides us with accurate results and enables us to investigate the phase diagram of those localized solutions within the critical regime, which goes far beyond previous results. We find that in this regime the phase diagram possesses a spiral structure adapting to the one recently found for non-uniform black strings. When approaching the common endpoint of both phases, the behavior of physical quantities is described by complex critical exponents giving rise to a discrete scaling symmetry. The numerically obtained values of the critical exponents agree remarkably well with those derived from the double-cone metric.

A fully discrete pseudospectral method for Fisher's equation on the whole line

In this paper, a fully discrete pseudospectral scheme for Fisher's equation whose solutions behave differently as x→+∞ and x→−∞ is presented using generalized Hermite interpolation. The convergence and the stability of the proposed scheme are analyzed. Numerical results coincide well with theoretical analysis and show efficiency of our algorithm.

Well Conditioned Pseudospectral Schemes with Tunable Basis for Fractional Delay Differential Equations

The main purpose of this work is to develop spectrally accurate and well conditioned pseudospectral schemes for solving fractional delay differential equations (FDDEs). The essential idea is to recast FDDEs into fractional integral equations (FIEs) and then discretize the FIEs via generalized fractional pseudospectral integration matrices (GFPIMs). We construct GFPIMs by employing the basis of weighted Lagrange interpolating functions, and provide an exact, efficient, and stable approach to computing GFPIMs. The GFPIM schemes have two remarkable features: (i) the endpoint singularity of the solution to FDDEs can be effectively captured via the tunable basis, and (ii) the linear system resulting from pseudospectral discretization is well conditioned. We also provide a rigorous convergence analysis for the particular FPIM schemes via a linear FIE with any $$\gamma >0$$ where $$\gamma $$ is the order of fractional integrals. Numerical results on benchmark FDDEs with smooth/singular solutions demonstrate the spectral rate of convergence for the GFPIM schemes. For FDDEs with piecewise smooth solutions, the GFPIM schemes can obtain accurate solutions but converge slowly due to their essential feature of “global” approximation on the entire time interval.

Optimal Control of Switched Systems by a Modified Pseudo Spectral Method

In the present paper, we develop a modified pseudospectral scheme for solving an optimal control problem which is governed by a switched dynamical system. Many real-world processes such as chemical processes, automotive systems and manufacturing processes can be modeled as such systems. For this purpose, we replace the problem with an alternative optimal control problem in which the switching times appear as unknown parameters. Using the Legendre-Gauss-Lobatto quadrature and the corresponding differentiation matrix, the alternative problem is discretized to a nonlinear programming problem. At last, we examine three examples in order to illustrate the efficiency of the proposed method.

Hermite Pseudospectral Method for the Time Fractional Diffusion Equation with Variable Coefficients

Abstract:We consider the initial value problem of the time fractional diffusion equation on the whole line and the fractional derivative is described in Caputo sense. A fully discrete Hermite pseudospectral approximation scheme is structured basing Hermite-Gauss points in space and finite difference in time. Unconditionally stability and convergence are proved. Numerical experiments are presented and the results conform to our theoretical analysis.

Efficient numerical schemes for the solution of generalized time fractional Burgers type equations

The main aim of this paper is to propose two semi-implicit Fourier pseudospectral schemes for the solution of generalized time fractional Burgers type equations, with an analysis of consistency, stability, and convergence. Under some assumptions, the unconditional stability of the schemes is shown. In implementation of these schemes, the fast Fourier transform (FFT) can be used efficiently to improve the computational cost. Various test problems are included to illustrate the results that we have obtained regarding the proposed schemes. The results of numerical experiments are compared with analytical solutions and other existing methods in the literature to show the efficiency of proposed schemes in both accuracy and CPU time. As numerical solution of fractional stochastic nonlinear partial differential equations driven by Brownian motions are among current related research interests, we report the performance of these schemes on stochastic time fractional Burgers equation as well.

Direct numerical simulation of scalar transport in turbulent flows over progressive surface waves

The transport of passive scalars in turbulent flows over progressive water waves is studied using direct numerical simulation. A combined pseudo-spectral and finite-difference scheme on a wave-surface-fitted grid is used to simulate the flow and scalar fields above the wave surface. Three representative wave ages (i.e. wave-to-wind speed ratios) are considered, corresponding to slow, intermediate and fast wind-waves, respectively. For each wave condition, four Schmidt numbers are considered for the scalar transport. The presence of progressive surface waves is found to induce significant wave-phase-correlated variation to the scalar field, with the phase dependence varying with the wave age. The time- and plane-averaged profiles of the scalar over waves of various ages exhibit similar vertical structures as those found in turbulence over a flat wall, but with the von Kármán constant and effective wave surface roughness for the mean scalar profile exhibiting considerable variation with the wave age. The profiles of the root-mean-square scalar fluctuations and the horizontal scalar flux exhibit good scaling in the viscous sublayer that agrees with the scaling laws previously reported for flat-wall turbulence, but with noticeable wave-induced variation in the viscous wall region. The profiles of the vertical scalar flux in the viscous sublayer exhibit apparent discrepancies from the reported scaling law for flat-wall turbulence, due to a negative vertical flux region above the windward face of the wave crest. Direct observation and quadrant-based conditional averages indicate that the wave-dependent distributions of the scalar fluctuations and fluxes are highly correlated with the coherent vortical structures in the turbulence, which exhibit clear wave-dependent characteristics in terms of both shape and preferential location.

Analysis of a Chebyshev-type pseudo-spectral scheme for the nonlinear Schrödinger equation

In this paper, we derive several error estimates that are pertinent to the study of Chebyshev-type spectral approximations on the real line. The results are applied to construct a stable and accurate pseudo-spectral Chebyshev scheme for the nonlinear Schrödinger equation. The new technique has several computational advantages as compared to Fourier and Hermite-type spectral schemes, described in the literature (see e.g., [1]–[3]. Similar to Hermite-type methods, we do not require domain truncation and/or use of artificial boundary conditions. At the same time, the computational complexity is comparable to the best Fourier-type spectral methods described in the literature.

A Regularized Newton Method for Computing Ground States of Bose–Einstein Condensates

In this paper, we propose a regularized Newton method for computing ground states of Bose-Einstein condensates (BECs), which can be formulated as an energy minimization problem with a spherical constraint. The energy functional and constraint are discretized by either the finite difference, or sine or Fourier pseudospectral discretization schemes and thus the original infinite dimensional nonconvex minimization problem is approximated by a finite dimensional constrained nonconvex minimization problem. Then an initial solution is first constructed by using a feasible gradient type method, which is an explicit scheme and maintains the spherical constraint automatically. To accelerate the convergence of the gradient type method, we approximate the energy functional by its second-order Taylor expansion with a regularized term at each Newton iteration and adopt a cascadic multigrid technique for selecting initial data. It leads to a standard trust-region subproblem and we solve it again by the feasible gradient type method. The convergence of the regularized Newton method is established by adjusting the regularization parameter as the standard trust-region strategy. Extensive numerical experiments on challenging examples, including a BEC in three dimensions with an optical lattice potential and rotating BECs in two dimensions with rapid rotation and strongly repulsive interaction, show that our method is efficient, accurate and robust.

Quantification of mixing in vesicle suspensions using numerical simulations in two dimensions.

We study mixing in Stokesian vesicle suspensions in two dimensions on a cylindrical Couette apparatus using numerical simulations. The vesicle flow simulation is done using a boundary integral method, and the advection-diffusion equation for the mixing of the solute is solved using a pseudo-spectral scheme. We study the effect of the area fraction, the viscosity contrast between the inside (the vesicles) and the outside (the bulk) fluid, the initial condition of the solute, and the mixing metric. We compare mixing in the suspension with mixing in the Couette apparatus without vesicles. On the one hand, the presence of vesicles in most cases slightly suppresses mixing. This is because the solute can be only diffused across the vesicle interface and not advected. On the other hand, there exist spatial distributions of the solute for which the unperturbed Couette flow completely fails to mix whereas the presence of vesicles enables mixing. We derive a simple condition that relates the velocity and solute and can be used to characterize the cases in which the presence of vesicles promotes mixing.

Rational Chebyshev pseudospectral method for long-short wave equations

We consider the initial boundary value problem of the Long-Short wave equations on the whole line. Firstly, a three level linear fully discrete pseudospectral scheme are structured based on central difference in time and rational Chebyshev functions in space which are orthogonal in the L2 space with weight 1. Secondly, the first-order differential matrix about rational Chebyshev functions is derived by the first-order differential matrix of Chebyshev polynomials, the entries of the matrix are just Chebyshev polynomials and Chebyshev Gauss collocation points. Thirdly, the numerical implementations are described and numerical results for the rational Chebyshev pseudospectral scheme are verified that a second-order accuracy in time and spectral accuracy in space.

An Accurate Space–Time Pseudospectral Method for Solving Nonlinear Multi-Dimensional Heat Transfer Problems

In this paper, we consider the numerical solution of the nonlinear one- and two-dimensional heat transfer problems subject to the given initial conditions and linear Robin boundary conditions. We propose a pseudospectral scheme in both time and spatial discretizations for these problems. The discretization processes are constructed through the multi-variate interpolation of the desired solutions in terms of Chebyshev Gauss Lobbato collocation points. Operational matrices of differentiation are constructed via the tensor products for speeding up of the proposed numerical algorithms’ implementation. Some test problems are provided and the numerical simulations are illustrated to show the spectral accuracy in both space and time of the suggested scheme.

Symplectic spatial integration schemes for systems of balance equations

A method to generate geometric pseudo-spectral spatial discretization schemes for hyperbolic or parabolic partial differential equations is presented. It applies to the spatial discretization of systems of conservation laws with boundary energy flows and/or distributed source terms. The symplecticity of the proposed spatial discretization schemes is defined with respect to the natural power pairing (form) used to define the port-Hamiltonian formulation for the considered systems of balance equations. The method is applied to the resistive diffusion model, a parabolic equation describing the plasma dynamics in tokamaks. A symplectic Galerkin scheme with Bessel conjugated bases is derived from the usual Galerkin method, using the proposed method. Besides the spectral and energetic properties expected from the symplecticity of the method, it is shown that more accurate approximation of eigenfunctions and reduced numerical oscillations result from this choice of conjugated approximation bases. Finally, the obtained numerical results are validated against experimental data from the tokamak Tore Supra facility.

Fractional Pseudospectral Schemes with Equivalence for Fractional Differential Equations

The main purpose of this work is to provide new fractional pseudospectral schemes with equivalence for solving fractional differential equations (FDEs). We develop differential and integral fractional pseudospectral schemes, and prove their equivalence from the distinctive perspective of the Caputo fractional Birkhoff interpolation with zero initial condition. We introduce the notion of fractional pseudospectral differentiation/integration matrices in terms of weighted Lagrange interpolating functions, and provide exact, efficient, and stable approaches to computing these matrices via the Lagrange interpolation and the Jacobi--Gauss quadrature. Numerical results on benchmark FDEs demonstrate the performance of the proposed pseudospectral schemes.

Rapid Prototyping of Pyramidal Structured Absorbers for Ultrasound

Acoustic measurements or ultrasonic testing can be strongly affected by reflections or echoes from test tank walls. In order to create a non-reflecting environment equivalent to infinite medium, a pyramidal structured absorber (PSA) can be used to coat the walls of an ultrasonic tank. In this work, we model an array of tetragonal pyramid ultrasonic wave absorbers. This model is based on two coupled first-order equations describing the stress and particle velocity within an isotropic medium. For absorbing media, the Kelvin-Voigt model of viscoelasticity is used. The equations are discretized in 2D using an efficient time-stepping pseudo-spectral scheme that takes in consideration both, the acoustic properties and attenuation characteristics of the composite materials. We then built a 3D printed PSA using a Stratasys Objet500 Connex 3D printer, which allows to combine photopolymers in specific concentrations and microstructures. We designed PSA covering the frequency ranges from 0.5 MHz to 5 MHz and from 1 MHz to 10 MHz, with double homogeneous layer: a core of rubber material with a skin of a variety of elastomers by combining rigid and flexible materials. Each single pyramid contains two major parts: the ground of the pyramid (9.4 mm base × 4.7 mm height, for 0.5 MHz and 4.7 mm base × 2.35 mm height, for 1 MHz) and the body of the pyramid (23.5 mm height, for 0.5 MHZ and 11.75 mm height, for 1 MHz). The measured echo-reduction was greater than 35 dB at the covering frequency range and the transmission loss was estimated by 20 dB. Echoes increase rapidly for frequencies below the minimum frequency of the covering range. The modeling and 3D printing of PSA with different sizes, in a wide range of frequencies, is a cost-effective custom solution for a wide range of applications including for example, radiation force balances, hydrophone mounts and medical ultrasound equipment.

Pseudospectral methods for density functional theory in bounded and unbounded domains

Classical Density Functional Theory (DFT) is a statistical–mechanical framework to analyse fluids, which accounts for nanoscale fluid inhomogeneities and non-local intermolecular interactions. DFT can be applied to a wide range of interfacial phenomena, as well as problems in adsorption, colloidal science and phase transitions in fluids. Typical DFT equations are highly non-linear, stiff and contain several convolution terms. We propose a novel, efficient pseudo-spectral collocation scheme for computing the non-local terms in real space with the help of a specialised Gauss quadrature. Due to the exponential accuracy of the quadrature and a convenient choice of collocation points near interfaces, we can use grids with a significantly lower number of nodes than most other reported methods. We demonstrate the capabilities of our numerical methodology by studying equilibrium and dynamic two-dimensional test cases with single- and multispecies hard-sphere and hard-disc particles modelled with fundamental measure theory, with and without van der Waals attractive forces, in bounded and unbounded physical domains. We show that our results satisfy statistical mechanical sum rules.

High-order IMEX-spectral schemes for computing the dynamics of systems of nonlinear Schrödinger/Gross–Pitaevskii equations

The aim of this paper is to build and validate some explicit high-order schemes, both in space and time, for simulating the dynamics of systems of nonlinear Schrödinger/Gross–Pitaevskii equations. The method is based on the combination of high-order IMplicit–EXplicit (IMEX) schemes in time and Fourier pseudo-spectral approximations in space. The resulting IMEXSP schemes are highly accurate, efficient and easy to implement. They are also robust when used in conjunction with an adaptive time stepping strategy and appear as an interesting alternative to time-splitting pseudo-spectral (TSSP) schemes. Finally, a complete numerical study is developed to investigate the properties of the IMEXSP schemes, in comparison with TSSP schemes, for one- and two-components systems of Gross–Pitaevskii equations.

On the ground states and dynamics of space fractional nonlinear Schrödinger/Gross–Pitaevskii equations with rotation term and nonlocal nonlinear interactions

In this paper, we propose some efficient and robust numerical methods to compute the ground states and dynamics of Fractional Schrödinger Equation (FSE) with a rotation term and nonlocal nonlinear interactions. In particular, a newly developed Gaussian-sum (GauSum) solver is used for the nonlocal interaction evaluation [31]. To compute the ground states, we integrate the preconditioned Krylov subspace pseudo-spectral method [4] and the GauSum solver. For the dynamics simulation, using the rotating Lagrangian coordinates transform [14], we first reformulate the FSE into a new equation without rotation. Then, a time-splitting pseudo-spectral scheme incorporated with the GauSum solver is proposed to simulate the new FSE. In parallel to the numerical schemes, we also prove some existence and nonexistence results for the ground states. Dynamical laws of some standard quantities, including the mass, energy, angular momentum and the center of mass, are stated. The ground states properties with respect to the fractional order and/or rotating frequencies, dynamics involving decoherence and turbulence together with some interesting phenomena are reported.

A directed tabu search method for solving controlled Volterra integral equations

This paper proposes a pseudo-spectral scheme for obtaining approximate optimal control and state. This computational scheme represents the solution of the optimal control problem (OCP) by an mth degree Lagrange interpolating polynomial, using Legendre nodes. Then, OCP of non-linear Volterra integral equation is transformed into an optimization problem. Directed tabu search (DTS) method is utilized to derive the solutions of the optimal control and state as well as the optimal value of the objective function. In the DTS method, two neighborhood-local search strategies based on Nelder–Mead method and adaptive pattern search are applied. In addition, a tabu list with anti-cycling rules, the so-called tabu regions and semi-tabu regions are used. In addition, diversification and intensification search schemes are employed. DTS is able to converge to the global optimum solutions of a set of numerical examples. Therefore, some good balance between the diversification and intensification ensures a faster and efficient convergence to get quality solutions. Numerical examples are provided which confirm the reliability and efficiency of the proposed method. Moreover, a comparison is made with optimal solutions obtained by the other numerical methods in the literature.