Pseudospectral Approximation Research Articles

An efficient Jacobi pseudospectral approximation for nonlinear complex generalized Zakharov system

In this paper, we derive an efficient spectral collocation algorithm to solve numerically the nonlinear complex generalized Zakharov system (GZS) subject to initial-boundary conditions. The Jacobi pseudospectral approximation is investigated for spatial approximation of the GZS. It possesses the spectral accuracy in space. The Jacobi–Gauss–Lobatto quadrature rule is established to treat the boundary conditions, and then the problem with its boundary conditions is reduced to a system of ordinary differential equations in time variable. This scheme has the advantage of allowing us to obtain the spectral solution in terms of the Jacobi parameters α and β, which therefore means that the algorithm holds a number of collocation methods as special cases. Finally, two illustrative examples are implemented to assess the efficiency and high accuracy of the Jacobi pseudo-spectral scheme.

New spectral techniques for systems of fractional differential equations using fractional-order generalized Laguerre orthogonal functions

Abstract Fractional-order generalized Laguerre functions (FGLFs) are proposed depends on the definition of generalized Laguerre polynomials. In addition, we derive a new formula expressing explicitly any Caputo fractional-order derivatives of FGLFs in terms of FGLFs themselves. We also propose a fractional-order generalized Laguerre tau technique in conjunction with the derived fractional-order derivative formula of FGLFs for solving Caputo type fractional differential equations (FDEs) of order ν (0 < ν < 1). The fractional-order generalized Laguerre pseudo-spectral approximation is investigated for solving nonlinear initial value problem of fractional order ν. The extension of the fractional-order generalized Laguerre pseudo-spectral method is given to solve systems of FDEs. We present the advantages of using the spectral schemes based on FGLFs and compare them with other methods. Several numerical example are implemented for FDEs and systems of FDEs including linear and nonlinear terms. We demonstrate the high accuracy and the efficiency of the proposed techniques.

A shifted Jacobi collocation algorithm for wave type equations with non-local conservation conditions

Abstract In this paper, we propose an efficient spectral collocation algorithm to solve numerically wave type equations subject to initial, boundary and non-local conservation conditions. The shifted Jacobi pseudospectral approximation is investigated for the discretization of the spatial variable of such equations. It possesses spectral accuracy in the spatial variable. The shifted Jacobi-Gauss-Lobatto (SJ-GL) quadrature rule is established for treating the non-local conservation conditions, and then the problem with its initial and non-local boundary conditions are reduced to a system of second-order ordinary differential equations in temporal variable. This system is solved by two-stage forth-order A-stable implicit RK scheme. Five numerical examples with comparisons are given. The computational results demonstrate that the proposed algorithm is more accurate than finite difference method, method of lines and spline collocation approach

GPELab, a Matlab toolbox to solve Gross–Pitaevskii equations I: Computation of stationary solutions

This paper presents GPELab (Gross–Pitaevskii Equation Laboratory), an advanced easy-to-use and flexible Matlab toolbox for numerically simulating many complex physics situations related to Bose–Einstein condensation. The model equation that GPELab solves is the Gross–Pitaevskii equation. The aim of this first part is to present the physical problems and the robust and accurate numerical schemes that are implemented for computing stationary solutions, to show a few computational examples and to explain how the basic GPELab functions work. Problems that can be solved include: 1d, 2d and 3d situations, general potentials, large classes of local and nonlocal nonlinearities, multi-components problems, and fast rotating gases. The toolbox is developed in such a way that other physics applications that require the numerical solution of general Schrödinger-type equations can be considered. Program summaryProgram title: GPELabCatalogue identifier: AETU_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AETU_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 26552No. of bytes in distributed program, including test data, etc.: 611289Distribution format: tar.gzProgramming language: Matlab.Computer: PC, Mac.Operating system: Windows, Mac OS, Linux.Has the code been vectorized or parallelized?: YesRAM: 4000 MegabytesClassification: 2.7, 4.6, 7.7.Nature of problem:Computing stationary solutions for a class of systems (multi-components) of Gross–Pitaevskii equations in 1d, 2d and 3d. This program is particularly well designed for the computation of ground states of Bose–Einstein condensates as well as dynamics.Solution method:We use the imaginary-time method with a Semi-Implicit Backward Euler scheme, a pseudo-spectral approximation and a Krylov subspace method.Running time:From a few minutes for simple problems to a week for more complex situations on a medium computer.

The operational matrix of Caputo fractional derivatives of modified generalized Laguerre polynomials and its applications

In this paper, the modified generalized Laguerre operational matrix (MGLOM) of Caputo fractional derivatives is constructed and implemented in combination with the spectral tau method for solving linear multi-term FDEs on the half-line. In this approach, truncated modified generalized Laguerre polynomials (MGLP) together with the modified generalized Laguerre operational matrix of Caputo fractional derivatives are analyzed and applied for numerical integration of such equations subject to initial conditions. The modified generalized Laguerre pseudo-spectral approximation based on the modified generalized Laguerre operational matrix is investigated to reduce the nonlinear multi-term FDEs and their initial conditions to a nonlinear algebraic system. Through some numerical experiments, we evaluate the accuracy and efficiency of the proposed methods. The methods are easy to implement and yield very accurate results.

Control and Synchronization of Neuron Ensembles

Synchronization of oscillations is a phenomenon prevalent in natural, social, and engineering systems. Controlling synchronization of oscillating systems is motivated by a wide range of applications from neurological treatment of Parkinson's disease to the design of neurocomputers. In this article, we study the control of an ensemble of uncoupled neuron oscillators described by phase models. We examine controllability of such a neuron ensemble for various phase models and, furthermore, study the related optimal control problems. In particular, by employing Pontryagin's maximum principle, we analytically derive optimal controls for spiking single- and two-neuron systems, and analyze the applicability of the latter to an ensemble system. Finally, we present a robust computational method for optimal control of spiking neurons based on pseudospectral approximations. The methodology developed here is universal to the control of general nonlinear phase oscillators.

Jaguar: A high‐performance quantum chemistry software program with strengths in life and materials sciences

Jaguar is anab initioquantum chemical program that specializes in fast electronic structure predictions for molecular systems of medium and large size. Jaguar focuses on computational methods with reasonable computational scaling with the size of the system, such as density functional theory (DFT) and local second‐order Møller–Plesset perturbation theory. The favorable scaling of the methods and the high efficiency of the program make it possible to conduct routine computations involving several thousand molecular orbitals. This performance is achieved through a utilization of the pseudospectral approximation and several levels of parallelization. The speed advantages are beneficial for applying Jaguar in biomolecular computational modeling. Additionally, owing to its superior wave function guess for transition‐metal‐containing systems, Jaguar finds applications in inorganic and bioinorganic chemistry. The emphasis on larger systems and transition metal elements paves the way toward developing Jaguar for its use in materials science modeling. The article describes the historical and new features of Jaguar, such as improved parallelization of many modules, innovations inab initiopKa prediction, and new semiempirical corrections for nondynamic correlation errors in DFT. Jaguar applications in drug discovery, materials science, force field parameterization, and other areas of computational research are reviewed. Timing benchmarks and other results obtained from the most recent Jaguar code are provided. The article concludes with a discussion of challenges and directions for future development of the program. © 2013 Wiley Periodicals, Inc.

Guidance and Control for Planetary Landing: Flatness-Based Approach

The guidance, navigation, and control problems for autonomous planetary entry descent and landing pose a number of interesting challenges. In this paper, the guidance and control problems for the descent phase are considered, and an approach based on the flatness property of the descent equations is proposed. A flat reformulation of the 2-D descent equations is presented and flatness-based analytical solutions and numerical schemes based on polynomial and pseudo-spectral approximations for the problem of guidance law design are compared. Subsequently, the problem of designing trajectory tracking control laws is considered and two design approaches are developed and compared in a simulation study. The first approach is based on feedforward linearization and leads to a nonlinear controller, while the second one is based on linear time-varying feedback computed by linearizing the descent equations along the computed trajectory. Finally, a simulation study is carried out, the results of which show that the nonlinear flatness-based controller provides a more satisfactory solution in terms of robustness.

Adaptive Smolyak Pseudospectral Approximations

Polynomial approximations of computationally intensive models are central to uncertainty quantification. This paper describes an adaptive method for nonintrusive pseudospectral approximation, based on Smolyak's algorithm with generalized sparse grids. We rigorously analyze and extend the nonadaptive method proposed in [P. G. Constantine, M. S. Eldred, and E. T. Phipps, Comput. Methods Appl. Mech. Engrg., 229--232 (2012), pp. 1--12], and compare it to a common alternative approach for using sparse grids to construct polynomial approximations, direct quadrature. Analysis of direct quadrature shows that $\mathcal{O}(1)$ errors are an intrinsic property of some configurations of the method, as a consequence of internal aliasing. We provide precise conditions, based on the chosen polynomial basis and quadrature rules, under which this aliasing error occurs. We then establish theoretical results on the accuracy of Smolyak pseudospectral approximation, and show that the Smolyak approximation avoids internal alia...

Two Efficient Generalized Laguerre Spectral Algorithms for Fractional Initial Value Problems

We present a direct solution technique for approximating linear multiterm fractional differential equations (FDEs) on semi-infinite interval, using generalized Laguerre polynomials. We derive the operational matrix of Caputo fractional derivative of the generalized Laguerre polynomials which is applied together with generalized Laguerre tau approximation for implementing a spectral solution of linear multiterm FDEs on semi-infinite interval subject to initial conditions. The generalized Laguerre pseudo-spectral approximation based on the generalized Laguerre operational matrix is investigated to reduce the nonlinear multiterm FDEs and its initial conditions to nonlinear algebraic system, thus greatly simplifying the problem. Through several numerical examples, we confirm the accuracy and performance of the proposed spectral algorithms. Indeed, the methods yield accurate results, and the exact solutions are achieved for some tested problems.

A New Legendre Spectral Galerkin and Pseudo-Spectral Approximations for Fractional Initial Value Problems

We extend the application of the Galerkin method for treating the multiterm fractional differential equations (FDEs) subject to initial conditions. A new shifted Legendre-Galerkin basis is constructed which satisfies exactly the homogeneous initial conditions by expanding the unknown variable using a new polynomial basis of functions which is built upon the shifted Legendre polynomials. A new spectral collocation approximation based on the Gauss-Lobatto quadrature nodes of shifted Legendre polynomials is investigated for solving the nonlinear multiterm FDEs. The main advantage of this approximation is that the solution is expanding by a truncated series of Legendre-Galerkin basis functions. Illustrative examples are presented to ensure the high accuracy and effectiveness of the proposed algorithms are discussed.

Stability and Convergence Analysis of Fully Discrete Fourier Collocation Spectral Method for 3-D Viscous Burgers’ Equation

This paper analyzes the stability and convergence of the Fourier pseudospectral method coupled with a variety of specially designed time-stepping methods of up to fourth order, for the numerical solution of a three dimensional viscous Burgers' equation. There are three main features to this work. The first is a lemma which provides for an L 2 and H 1 bound on a nonlinear term of polynomial type, despite the presence of aliasing error. The second feature of this work is the development of stable time-stepping methods of up to fourth order for use with pseudospectral approximations of the three dimensional viscous Burgers' equation. Finally, the main result in this work is that the pseudospectral method coupled with the carefully designed time-discretizations is stable provided only that the time-step and spatial grid-size are bounded by two constants over a finite time. It is notable that this stability condition does not impose a restriction on the time-step that is dependent on the spatial grid size, a fact that is especially useful for three dimensional simulations.

A trigonometric integrator pseudospectral discretization for the N-coupled nonlinear Klein–Gordon equations

A scheme, stemming from the use of pseudospectral approximations to spatial derivatives followed by a time integrator based on trigonometric polynomials, is proposed for the numerical solutions of the N-coupled nonlinear Klein–Gordon equations. Numerical tests on one- and three-coupled Klein–Gordon equations are presented, which are geared towards understanding the accuracy and stability, and illustrating its efficiency and high resolution capacity in applications.

Sparse pseudospectral approximation method

Multivariate global polynomial approximations – such as polynomial chaos or stochastic collocation methods – are now in widespread use for sensitivity analysis and uncertainty quantification. The pseudospectral variety of these methods uses a numerical integration rule to approximate the Fourier-type coefficients of a truncated expansion in orthogonal polynomials. For problems in more than two or three dimensions, a sparse grid numerical integration rule offers accuracy with a smaller node set compared to tensor product approximation. However, when using a sparse rule to approximately integrate these coefficients, one often finds unacceptable errors in the coefficients associated with higher degree polynomials.By reexamining Smolyak’s algorithm and exploiting the connections between interpolation and projection in tensor product spaces, we construct a sparse pseudospectral approximation method that accurately reproduces the coefficients for basis functions that naturally correspond to the sparse grid integration rule. The compelling numerical results show that this is the proper way to use sparse grid integration rules for pseudospectral approximation.

Uniform convergence of the legendre spectral method for the Zakharov equations

The Legendre spectral and pseudospectral approximations are proposed for the standard Zakharov equations with initial boundary conditions. Optimal H1 error estimate of the method is given for both semidiscrete and fully discrete schemes. The uniform convergence for the parameter e relative to the acoustic speed is proved. Moreover, the multidomain Legendre spectral scheme is also constructed, which can be implemented in parallel. Finally, numerical results in single domain and multidomain verify the high accuracy of the Legendre spectral method. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013

Optimal Control of Inhomogeneous Ensembles

Inhomogeneity, in its many forms, appears frequently in practical physical systems. Readily apparent in quantum systems, inhomogeneity is caused by hardware imperfections, measurement inaccuracies, and environmental variations, and subsequently limits the performance and efficiency achievable in current experiments. In this paper, we provide a systematic methodology to mathematically characterize and optimally manipulate inhomogeneous ensembles with concepts taken from ensemble control. In particular, we develop a computational method to solve practical quantum pulse design problems cast as optimal ensemble control problems, based on multidimensional pseudospectral approximations. We motivate the utility of this method by designing pulses for both standard and novel applications. We also show the convergence of the pseudospectral method for optimal ensemble control. The concepts developed here are applicable beyond quantum control, such as to neuron systems, and furthermore to systems with by parameter uncertainty, which pervade all areas of science and engineering.

Multidomain pseudospectral methods for nonlinear convection-diffusion equations

Multidomain pseudospectral approximations to nonlinear convection-diffusion equations are considered. The schemes are formulated with the Legendre-Galerkin method, but the nonlinear term is collocated at the Legendre/Chebyshev-Gauss-Lobatto points inside each subinterval. Appropriate base functions are introduced so that the matrix of the system is sparse, and the method can be implemented efficiently and in parallel. The stability and the optimal rate of convergence of the methods are proved. Numerical results are given for both the single domain and the multidomain methods to make a comparison.

Hermite spectral and pseudospectral methods for numerical differentiation

A numerical differentiation problem for a given function with noisy data is discussed in this paper. A mollification method based on spanned by Hermite functions is proposed and the mollification parameter is chosen by a discrepancy principle. The convergence estimates of the derivatives are obtained. To get a practical approach, we also derive corresponding results for pseudospectral (Hermite–Gauss interpolation) approximations. Numerical examples are given to show the efficiency of the method.

Analysis and comparison of numerical methods for the Klein–Gordon equation in the nonrelativistic limit regime

We analyze rigourously error estimates and compare numerically temporal/spatial resolution of various numerical methods for solving the Klein–Gordon (KG) equation in the nonrelativistic limit regime, involving a small parameter $${0 < {\varepsilon}\ll 1}$$ which is inversely proportional to the speed of light. In this regime, the solution is highly oscillating in time, i.e. there are propagating waves with wavelength of $${O({\varepsilon}^2)}$$ and O(1) in time and space, respectively. We begin with four frequently used finite difference time domain (FDTD) methods and obtain their rigorous error estimates in the nonrelativistic limit regime by paying particularly attention to how error bounds depend explicitly on mesh size h and time step τ as well as the small parameter $${{\varepsilon}}$$. Based on the error bounds, in order to compute ‘correct’ solutions when $${0 < {\varepsilon}\ll1}$$, the four FDTD methods share the same $${{\varepsilon}}$$ -scalability: $${\tau=O({\varepsilon}^3)}$$. Then we propose new numerical methods by using either Fourier pseudospectral or finite difference approximation for spatial derivatives combined with the Gautschi-type exponential integrator for temporal derivatives to discretize the KG equation. The new methods are unconditionally stable and their $${{\varepsilon}}$$ -scalability is improved to τ = O(1) and $${\tau=O({\varepsilon}^2)}$$ for linear and nonlinear KG equations, respectively, when $${0 < {\varepsilon}\ll1}$$. Numerical results are reported to support our error estimates.

Large-wave simulation of three-dimensional, cross-shore and oblique, spilling breaking on constant slope beach

In this work, the large-wave simulation (LWS) method is adapted for application in spilling wave breaking over a constant slope beach. According to LWS, large scales of velocities, pressure and free-surface elevation are numerically resolved, while the corresponding unresolved scale effects are taken into consideration by a subgrid scale (SGS) model for wave and eddy stresses. The model may be not fully applicable in very shallow water, close to the shoreline, where the unresolved, turbulent, free-surface oscillation is of the same order with the water depth. Time integration of the Euler equations is achieved by a two-stage fractional scheme, combined with a hybrid scheme for spatial discretization, consisting of finite difference and pseudospectral approximation methods. Model parameters are calibrated by comparison to available experimental data of free-surface elevation and velocities in the surf zone for cross-shore incoming waves. The action of the wave SGS stresses in the outer coastal and surf zones initiates breaking and generates appropriate vorticity, in the form of an eddy structure (surface roller), at the breaking wavefront. At incipient breaking, both advection and gravity contribute to the vorticity flux at the free surface, while only after the full development of the surface roller, the effect of advection becomes stronger. The SGS model is also utilized to simulate propagation, refraction and breaking of oblique incoming waves. The gradual breaking and dissipation of wave crestlines and the surface roller structure along the breaking wavefront are automatically captured without any empirical input, such as data for the roller shape or the wave propagation angle at breaking.