Pseudospectral Discretization Research Articles

A direct method for the Boltzmann equation based on a pseudo-spectral velocity space discretization

A deterministic method is proposed for solving the Boltzmann equation. The method employs a Galerkin discretization of the velocity space and adopts, as trial and test functions, the collocation basis functions based on weights and roots of a Gauss–Hermite quadrature. This is defined by means of half- and/or full-range Hermite polynomials depending whether or not the distribution function presents a discontinuity in the velocity space. The resulting semi-discrete Boltzmann equation is in the form of a system of hyperbolic partial differential equations whose solution can be obtained by standard numerical approaches. The spectral rate of convergence of the results in the velocity space is shown by solving the spatially uniform homogeneous relaxation to equilibrium of Maxwell molecules. As an application, the two-dimensional cavity flow of a gas composed of hard-sphere molecules is studied for different Knudsen and Mach numbers. Although computationally demanding, the proposed method turns out to be an effective tool for studying subsonic slightly rarefied gas flows.

Fast accurate computation of the fully nonlinear solitary surface gravity waves

In this short note, we present an easy to implement and fast algorithm for the computation of the steady solitary gravity wave solution of the free surface Euler equations in irrotational motion. First, the problem is reformulated in a fixed domain using the conformal mapping technique. Second, the problem is reduced to a single equation for the free surface. Third, this equation is solved using Petviashvili’s iterations together with pseudo-spectral discretisation. This method has a super-linear complexity, since the most demanding operations can be performed using a FFT algorithm. Moreover, when this algorithm is combined with the multi-precision floating point computations, the results can be obtained to any arbitrary accuracy.

CONVERGENCE ANALYSIS OF THE PSEUDOSPECTRAL METHOD FOR LINEAR DAEs OF INDEX-2

This paper is concerned with the study of pseudospectral discretizations of the differential-algebraic equations (DAEs). Pseudospectral method based on Chebyshev polynomials are used to transcribe a given DAE into a system of algebraic equations. A posteriori error bound between the desired solution to the index 2, DAEs and the pseudospectral approximate solution of the problem is estimated in the weighted L2 norm. Some numerical experiments are considered to demonstrate the efficiency and the applicability of the method.

A stochastic pseudospectral and T-matrix algorithm for acoustic scattering by a class of multiple particle configurations

We consider absorption and scattering of acoustic waves from uncertain configurations comprising multiple two dimensional bodies with various material properties (sound-soft, sound-hard, absorbing and penetrable) and develop tools to address the problem of quantifying uncertainties in the acoustic cross sections of the configurations. The uncertainty arises because the locations and orientations of the particles in the configurations are described through random variables, and statistical moments of the far-fields induced by the stochastic configurations facilitate quantification of the uncertainty. We develop an efficient algorithm, based on a hybrid of the stochastic pseudospectral discretization (to truncate the infinite dimensional stochastic process) and an efficient stable truncated version of Waterman's T-matrix approach (for cost effective realization at each multiple particle configuration corresponding to the pseudospectral quadrature points) to simulate the statistical properties of the stochastic model. We demonstrate the efficiency of the algorithm for configurations with non-smooth and non-convex bodies with distinct material properties, and random locations and orientations with normal and log-normal distributions.

An Exponential Wave Integrator Sine Pseudospectral Method for the Klein--Gordon--Zakharov System

An exponential wave integrator sine pseudospectral method is presented and analyzed for discretizing the Klein--Gordon--Zakharov (KGZ) system with two dimensionless parameters $0<\varepsilon\le 1$ and $0<\gamma\le 1$ which are inversely proportional to the plasma frequency and the speed of sound, respectively. The main idea in the numerical method is to apply the sine pseudospectral discretization for spatial derivatives followed by using an exponential wave integrator for temporal derivatives in phase space. The method is explicit, symmetric in time, and it is of spectral accuracy in space and second-order accuracy in time for any fixed $\varepsilon=\varepsilon_0$ and $\gamma=\gamma_0$. In the $\mathcal{O}(1)$-plasma frequency and speed of sound regime, i.e., $\varepsilon=\mathcal{O}(1)$ and $\gamma=\mathcal{O}(1)$, we establish rigorously error estimates for the numerical method in the energy space $H^1\times L^2$. We also study numerically the resolution of the method in the simultaneous high-plasma-frequency and subsonic limit regime, i.e., $(\varepsilon,\gamma)\to 0$ under $\varepsilon\lesssim\gamma$. In fact, in this singular limit regime, the solution of the KGZ system is highly oscillating in time, i.e., there are propagating waves with wavelength of $\mathcal{O}(\varepsilon^2)$ and $\mathcal{O}(1)$ in time and space, respectively. Our extensive numerical results suggest that, in order to compute “correct” solutions in the simultaneous high-plasma-frequency and subsonic limit regime, the meshing strategy (or $\varepsilon$-scalability) is time step $\tau=\mathcal{O}(\varepsilon^2)$ and mesh size $h=\mathcal{O}(1)$ independent of $\varepsilon$. Finally, we also observe numerically that the method has the property of near conservation of the energy over long time in practical computations.

Discrete conservation laws and the convergence of long time simulations of the mkdv equation

Pseudospectral collocation methods and finite difference methods have been used for approximating an important family of soliton like solutions of the mKdV equation. These solutions present a structural instability which make difficult to approximate their evolution in long time intervals with enough accuracy. The standard numerical methods do not guarantee the convergence to the proper solution of the initial value problem and often fail by approaching solutions associated to different initial conditions. In this frame the numerical schemes that preserve the discrete invariants related to some conservation laws of this equation produce better results than the methods which only take care of a high consistency order. Pseudospectral spatial discretization appear as the most robust of the numerical methods, but finite difference schemes are useful in order to analyze the rule played by the conservation of the invariants in the convergence.

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.

Pseudospectral methods for Boussinesq-type equations in an annular domain with applications to mid-sized lakes

We present a numerical solution procedure for a two-dimensional Boussinesq-type shallow water model on annular domains using pseudospectral discretization methods, including practical implementation details such as spectral filtering to prevent aliasing-driven instabilities and efficient numerical linear algebra techniques. The numerical model's potential for predicting and simulating wave motions in mid-sized lakes is illustrated with three test cases: (1) wave diffraction around an island and near-shore focusing; (2) the formation, propagation and destruction of wave trains and solitary-like waves in rotating basins; and (3) the influence of bottom bathymetry on wave formation and propagation from a Kelvin-seiche.

Manifestly gauge invariant discretizations of the Schrödinger equation

Grid-based discretizations of the time dependent Schrödinger equation coupled to an external magnetic field are converted to manifest gauge invariant discretizations. This is done using generalizations of ideas used in classical lattice gauge theory, and the process defined is applicable to a large class of discretized differential operators. In particular, popular discretizations such as pseudospectral discretizations using the fast Fourier transform can be transformed to gauge invariant schemes. Also generic gauge invariant versions of generic time integration methods are considered, enabling completely gauge invariant calculations of the time dependent Schrödinger equation. Numerical examples illuminating the differences between a gauge invariant discretization and conventional discretization procedures are also presented.

A fourth-order split-step pseudospectral scheme for the Kuramoto–Tsuzuki equation

The numerics of the Kuramoto–Tsuzuki equation is dealt with in this paper. We propose a split-step Fourier pseudospectral discretization for solving the problem, which is split into one linear subproblem and one nonlinear subproblem. The nonlinear subproblem is integrated exactly via solving the equations for the amplitude and phase angle of the unknown complex-valued function respectively. The linear subproblem is first approximated by Fourier pseudospectral discretization to the spatial derivative, and then integrated exactly in phase space via solving the equations for the Fourier coefficients analytically. We apply a fourth-order splitting integration in time advances, and therefore the overall error in space discretization is of spectral order and the overall error in time discretization is of fourth order which merely comes from the splitting. The scheme is fully explicit, easy to implement and quite efficient thanks to FFT. Moreover, it is time reversible and gauge invariant which are two properties in the continuous problem. Extensive numerical results are reported, which are geared towards testing the convergence and demonstrating the efficiency and accuracy.

Influence of waves on Lagrangian acceleration in two-dimensional turbulent flows

The Lagrangian statistics in rotating Saint-Venant turbulence are studied by means of direct numerical simulation using a pseudo-spectral discretization fully resolving, both in time and space, all the inertio-gravity waves present in the system. To understand the influence of waves, three initial conditions are considered, one which is dominated by waves, one which is dominated by vortices, and one which is intermediate between these two extreme cases. It is found that Lagrangian acceleration is strongly influenced by the presence of waves, as are the Lagrangian velocity increments, for incremements that are sufficiently short, and that vortexmotion influences mostly the component of Lagrangian acceleration which is perpendicular to the local velocity.

A short note on simplified pseudospectral methods for computing ground state and dynamics of spherically symmetric Schrödinger–Poisson–Slater system

In a recent paper we proposed and compared various approaches to compute the ground state and dynamics of the Schrödinger–Poisson–Slater (SPS) system for general external potential and initial condition, concluding that the methods based on sine pseudospectral discretization in space are the best candidates. This note is concerned with the case that the external potential and initial condition are spherically symmetric. For the SPS system with spherical symmetry, via applying a proper change of variables into the reduced quasi-1D model we simplify the methods proposed for the general 3D case such that both the memory and computational load are significantly reduced.

Calculations of low-lying states of hydrogen atom in strong magnetic field by generalized pseudospectral method

A very simple and highly accurate method of investigating the behavior of atomic system in a strong magnetic field has been developed based on the generalized pseudospectral discretization. As examples, we present the calculations of the binding energies and quadruple moments for the ground and low-lying excited states of the hydrogen atom in a magnetic field ranging from zero up to 1000 a.u.. The obtained results are found to be in excellent agreement with other high-accuracy theoretical calculations. The present method may be straightforwardly applied to cross electric-magnetic field in an arbitrary orientation.

Error-landscape-based multiobjective calibration of the Smagorinsky eddy-viscosity using high-Reynolds-number decaying turbulence data

A suite of large-eddy simulations (LESs) of decaying homogeneous isotropic turbulence at high Reynolds numbers is performed and compared to wind-tunnel experiments in the tradition of Comte-Bellot and Corrsin. The error-landscape approach is used for the evaluation of the Smagorinsky model, and the results are used to identify an optimal combination of model parameter and resolution in a statistically robust fashion. The use of experimental reference data in the error-landscape approach allows to evaluate the optimal Smagorinsky coefficient at high Reynolds numbers and to perform detailed comparisons with analytical predictions. We demonstrate, using a pseudospectral discretization, that the optimal so-called Smagorinsky trajectory obtained from the error-landscape analysis converges at high simulation resolutions to the high-Re theoretical Lilly prediction for the Smagorinsky coefficient. Using modified wavenumbers in the same spectral code, the current study also presents error-landscape results based on LES with “second-order” discretization errors. By slightly revising Lilly’s analysis, we show that including the effect of numerical discretization when evaluating the strain-rate tensor needed in the subgrid-scale model leads to a good prediction of the optimal Smagorinsky parameter obtained from the corresponding error-landscape. Using similar analytical tools, we further demonstrate that the dynamic procedure can also be adapted to better account for the effects of discretization and test-filter shape.

Analytical Solutions to Spacecraft Formation-Flying Guidance Using Virtual Motion Camouflage

Formation-flying has been recognized as one of the enabling technologies for a variety of space mission concepts because of the potential benefits in cost reduction, enhanced flexibility, increased observational baseline, and better survivability and reliability. One important requirement of the formation-flying system is the ability to generate trajectories in real time reliably to maximize mission return. This paper presents simple and analytical open-loop guidance solutions to formation-flying reconfiguration trajectory design problems through the bioinspired virtual motion camouflage methodology and the pseudospectral discretization approach. The algorithms investigated here can be applied to different kinds of initial and final conditions as well as different linearized models considering the eccentricity effects and/or the J 2 perturbation. Simulations are used to demonstrate the capabilities of the achieved analytical solutions for two formation-flying reconfiguration scenarios.

On the numerical solution of space–time fractional diffusion models

A flexible numerical scheme for the discretization of the space–time fractional diffusion equation is presented. The model solution is discretized in time with a pseudo-spectral expansion of Mittag–Leffler functions. For the space discretization, the proposed scheme can accommodate either low-order finite-difference and finite-element discretizations or high-order pseudo-spectral discretizations. A number of examples of numerical solutions of the space–time fractional diffusion equation are presented with various combinations of the time and space derivatives. The proposed numerical scheme is shown to be both efficient and flexible.

Time-dependent generalized pseudospectral method for accurate treatment of multiphoton processes of diatomic molecules in intense laser fields

We present a numerical method based on the generalized pseudospectral discretization in prolate spheroidal coordinates and split-operator time propagation. The method has been applied for the study of multiphoton processes of H + 2 and N 2 diatomic molecules in intense laser fields. It proves very accurate while using only moderate computer resources.

Computation of Asymptotic Stability for a Class of Partial Differential Equations with Delay

In this paper the question of asymptotic stability for retarded functional reaction diffusion equations is faced. Due to the infinite dimension of the problem a numerical approach is necessary. Here we propose a technique based on a pseudospectral discretization in time and on a spectral discretization in space of the infinitesimal generator associated to the semigroup of solution operators. Some numerical experiments on a Hutchinson-type equation modeling heat conduction in a rod with spatially variable gain delayed feedback are performed to show the efficiency of the scheme. This work represents a nontrivial extension of previous work of the authors on the computation of asymptotic stability for delay differential equations.

Um tutorial sobre métodos pseudo-espectrais para controle ótimo computacional

Este artigo é um tutorial introdutório sobre controle ótimo pseudo-espectral. Em métodos pseudo-espectrais, uma função é aproximada como uma combinação linear de funções de base suaves, tipicamente escolhidas como polinômios de Legendre ou Chebyshev. A colocação de equações algébrico-diferenciais é realizada em pontos de colocação ortogonal, que são selecionados de modo a minimizar o erro de interpolação. Métodos pseudo-espectrais discretizam o problema de controle ótimo original de modo a convertê-lo em um problema de programação não-linear. Um otimizador numérico é então empregado para obter soluções localmente ótimas. Este artigo também descreve sucintamente a funcionalidade e a implementação de um pacote computacional de código aberto escrito em C++ chamado PSOPT. Tal pacote emprega métodos de discretização pseudo-spectrais para resolver problemas de controle ótimo com múltiplas fase. O PSOPT permite a utilização de métodos de Legendre ou Chebyshev, e possui características úteis tais como diferenciação automática, detecção de esparsidade e escalonamento automático. O uso de métodos pseudo-espectrais é ilustrado em dois problemas retirados da literatura de controle ótimo computacional.

Comparisons between sine-Gordon and perturbed nonlinear Schrödinger equations for modeling light bullets beyond critical collapse

The sine-Gordon (SG) equation and perturbed nonlinear Schrödinger (NLS) equations are studied numerically for modeling the propagation of two space dimensional (2D) localized pulses (the so-called light bullets) in nonlinear dispersive optical media. We begin with the (2 + 1) SG equation obtained as an asymptotic reduction in the two level dissipationless Maxwell–Bloch system, followed by the review on the perturbed NLS equation in 2D for SG pulse envelopes, which is globally well posed and has all the relevant higher order terms to regularize the collapse of standard critical (cubic focusing) NLS. The perturbed NLS is approximated by truncating the nonlinearity into finite higher order terms undergoing focusing–defocusing cycles. Efficient semi-implicit sine pseudospectral discretizations for SG and perturbed NLS are proposed with rigorous error estimates. Numerical comparison results between light bullet solutions of SG and perturbed NLS as well as critical NLS are reported, which validate that the solution of the perturbed NLS as well as its finite-term truncations are in qualitative and quantitative agreement with the solution of SG for the light bullets propagation even after the critical collapse of cubic focusing NLS. In contrast, standard critical NLS is in qualitative agreement with SG only before its collapse. As a benefit of such observations, pulse propagations are studied via solving the perturbed NLS truncated by reasonably many nonlinear terms, which is a much cheaper task than solving SG equation directly.