Fourier Pseudospectral Method Research Articles

Direct/split invariant-preserving Fourier pseudo-spectral methods for the rotation-two-component Camassa–Holm system with peakon solitons

The Fourier pseudo-spectral method is well suited to solve PDEs under the periodic boundary condition due to its high-order accuracy and easy-to-implement feature. In this paper, we explore as well as comparatively study four classes of Fourier pseudo-spectral schemes for solving the rotation-two-component Camassa–Holm system which possibly owns peakon solitons. Via exploiting inherent structural properties of the system, we reformulate it into two kinds of different equivalent forms and then apply the Fourier pseudo-spectral method to derive two spatial semi-discrete systems, both of which are proved to preserve the corresponding invariants including mass, momentum and energy. Subsequently, we construct two linearly implicit schemes based on Strang splitting technique and two nonlinear schemes, respectively, for both semi-discrete systems. Owing to the different equivalent forms in the structure, one of the nonlinear schemes preserves discrete mass and momentum, while the other one is shown to preserve all three invariants. Numerical results under the situation of smooth/nonsmooth initial values are provided for distinct types of solutions to test the accuracy in long time simulation and to verify the capacity of predicting water wave propagation, as well as advantages in preserving these invariants. For instance, the present schemes are shown to be at least 14 significant digits, improving upon 10 from ones in previous references.

Improved uniform error bounds on an exponential wave integrator method for the nonlinear Schrödinger equation with wave operator and weak nonlinearity

In this paper, we establish the improved uniform error bounds for the Lawson-type exponential wave integrator Fourier pseudo-spectral (LEWIFP) method for the nonlinear Schrödinger equation with wave operator (NLSW) and weak nonlinearity characterized by a small constant ε∈(0,1]. We first convert the NLSW to a coupled system and then consider the LEWIFP method for the coupled system. The LEWIFP method is proved to be time symmetric and mass-conservative which is very important from the point of view of numerical geometric integration. Through careful and rigorous convergence analysis, we establish improved uniform error bounds for the full-discretization at O(hm+ε2pτ2) in the long-time domain up to O(1/ε2p) where m is determined by the regularity conditions. These error bounds are much better than the classical error bounds O(hm+τ2) provided by the traditional error analysis. In error analysis, in addition to the classical tools such as mathematical induction and energy method, we adopt the regularity compensation oscillation (RCO) technique to analyze the accumulation of errors carefully. The numerical experiments support our error estimates and demonstrate the discrete mass-conservation. In addition, the numerical results show that the discrete energy is stable in the time domain up to O(1/ε2p).

Numerical Simulation of Flows Using the Fourier Pseudospectral Method and the Immersed Boundary Method

The present work proposes the application of a computational methodology based on the coupling of the Fourier Pseudospectral Method (FPSM) and the Immersed Boundary Method (IBM) for conducting flow simulations over slender airfoils. This methodology, termed IMERSPEC, leverages the benefits of both high accuracy and low computational cost inherent in pseudospectral methods, thanks to the utilization of the Fast Fourier Transform algorithm. IBM is employed to impose non-periodic boundary conditions in the Navier–Stokes equations, addressing the requirement of periodicity at boundaries for FPSM convergence and to accurately represent the immersed slender airfoil in the flow. The aerodynamic behavior of the analyzed profiles was assessed by calculating lift and drag coefficients, which were then compared with existing literature results. Consistently favorable outcomes were observed, particularly in flows at lower Reynolds numbers, demonstrating the effectiveness of the IMERSPEC methodology for simulating complex flows computationally. Additionally, weight functions, fundamental to IBM, are employed flexibly for aerodynamic force calculations. Specifically, within the same simulation, a Cubic function is utilized for drag calculation while a Hat function is employed for lift calculation, yielding results more closely aligned with the literature’s findings. This approach offers an alternative to previously proposed methods for IBM implementation.

An exact in time Fourier pseudospectral method with multiple conservation laws for three-dimensional Maxwell’s equations

Maxwell’s equations describe the propagation of electromagnetic waves and are therefore fundamental to understanding many problems encountered in the study of antennas and electromagnetics. The aim of this paper is to propose and analyse an efficient fully discrete scheme for solving three-dimensional Maxwell’s equations. This is accomplished by combining Fourier pseudospectral methods in space and exact formulation in time. Fast computation is efficiently implemented in the scheme by using the matrix diagonalisation method and fast Fourier transform algorithm which are well known in scientific computations. An optimal error estimate which is not encumbered by the CFL condition is established and the resulting scheme is proved to be of spectral accuracy in space and exact in time. Furthermore, the scheme is shown to have multiple conservation laws including discrete energy, helicity, momentum, symplecticity, and divergence-free field conservations. All the theoretical results of the accuracy and conservations are numerically illustrated by two numerical tests.

Coupling of the immersed boundary and Fourier pseudo-spectral methods applied to solve fluid–structure interaction problems

The current manuscript addresses fluid–structure interaction (FSI) problems to the analysis of vortex-induced vibration (VIV), employing the methodology designated as IMERSPEC2D. The approach seamlessly integrates the Fourier pseudo-spectral method (FPSM) and the immersed boundary method (IBM) to solve the Navier–Stokes (NS) and continuity equations. Simultaneously, the effects of cylindrical structure, involving 1 and 2 degrees of freedom (d.o.f.) are simulated, by incorporating structural motion equations into NS via the IBM source term and implementing a two-way FSI algorithm. In the paper are presented the comparison of low and high-order temporal integration methods applied to solve the dynamic behavior of the structure and the dimensionless of the cylinder’s structural motion equations, resulting in an increase in computational time step from 1E(-9\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-9$$\\end{document}) to 1E(-3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-3$$\\end{document}). Validation results to refined mesh simulations include the accurate representation of the lock-in phenomenon to simulations of 1 d.o.f. obtaining the range between reduced velocity 5.456 and 5.568 and the maximal amplitude is 0.352. Notably, the eight-shape pattern of the 2 d.o.f. cylinder displacement is obtained and the formation of C2S wake patterns is achieved, as presented by other authors. In conclusion, the dimensionless approach reduces computational time, while the high-order both of IMERSPEC methodology and time advancement integration to FSI improve the results of the cylinder’s motion and alters distinct vortex shedding patterns in fluid dynamics.

Highly efficient, robust and unconditionally energy stable second order schemes for approximating the Cahn-Hilliard-Brinkman system

In this paper, we present an efficient, robust, and unconditionally energy stable second-order scheme for solving the Cahn-Hilliard-Brinkman (CHB) model, which mathematically describes multiphase flow in porous media. Solving the CHB model is significantly challenging due to its high coupling and nonlinearity. Here, we utilize the scalar auxiliary variable (SAV) method to handle the nonlinear term in the Cahn-Hilliard system, followed by decoupling the model through the implicit-explicit (IMEX) Crank-Nicolson approach. Our constructed scheme employs an adaptive time step, enhancing the robustness of the scheme compared to a fixed time step. The unconditional energy stability of our scheme that integrates artificial stabilized term, is assured easily owing to the SAV method and is theoretically proven in the paper. Finally, numerical experiments employing the Fourier pseudo-spectral method to discrete the spatial variables were conducted in two-dimensional space to validate the numerical accuracy and efficiency of the proposed approach.

A class of arbitrarily high-order energy-preserving method for nonlinear Klein–Gordon–Schrödinger equations

In this paper, we develop a class of arbitrarily high-order energy-preserving time integrators for the nonlinear Klein–Gordon–Schrödinger equations. We employ Fourier pseudo-spectral method for spatial discretization, resulting in a semi-discrete system. Subsequently, we employ the Petrov-Galerkin method in time to obtain a fully-discrete system. We rigorously demonstrate that the proposed scheme preserves the original energy of the target system. Furthermore, we have proved that the mass of the system is also approximately preserved. To assess the accuracy of our method, we provide a simple estimate of the local error, revealing that the proposed approach achieves a temporal order of 2s. Additionally, we extend the proposed methods to the damped Klein–Gordon–Schrödinger equations, and the corresponding fully-discrete scheme preserves the original energy dissipation law of the system. We present numerical examples to verify the accuracy and robustness of the proposed scheme.

Uniform Error Bounds of an Energy-Preserving Exponential Wave Integrator Fourier Pseudo-Spectral Method for the Nonlinear Schrödinger Equation with Wave Operator and Weak Nonlinearity

Uniform Error Bounds of an Energy-Preserving Exponential Wave Integrator Fourier Pseudo-Spectral Method for the Nonlinear Schrödinger Equation with Wave Operator and Weak Nonlinearity

Explicit and structure-preserving exponential wave integrator Fourier pseudo-spectral methods for the Dirac equation in the simultaneously massless and nonrelativistic regime

Explicit and structure-preserving exponential wave integrator Fourier pseudo-spectral methods for the Dirac equation in the simultaneously massless and nonrelativistic regime

Improved uniform error estimates for the two-dimensional nonlinear space fractional Dirac equation with small potentials over long-time dynamics

We develop improved uniform error bounds on a second-order Strang splitting method for the long-time dynamics of the nonlinear space fractional Dirac equation (NSFDE) in two dimension (2D) with small electromagnetic potentials. First, a Strang splitting approach is implemented to discretize NSFDE in time. Afterwards the Fourier pseudospectral method is used to complete the discretization of NSFDE in space. With the aid of a second-order Strang splitting approach employed to the Dirac equation, the major local truncation error of the indicated numerical methods is established. Moreover, for the semi-discrete scheme and full-discretization, we rigorously demonstrate the improved, sharp uniform error estimates are O(ετ2) and O(h1m+h2m+ετ2) in virtue of the regularity compensation oscillation (RCO) technique. In the formulations, τ is the time step, hi(i=1,2) stands for spatial sizes in xi-directions, m is dependent on the regularity of solutions, and ε∈(0,1]. In order to verify our error bounds and to illustrate some fascinating long-time dynamical behaviors of the NSFDE with honeycomb lattice potentials for varied ε, numerical investigations are presented.

Ground states of helicoidal spin-orbit-coupled spin-1 bosons

The ground states of spin-1 Bose–Einstein condensate under helicoidal spin–orbit-coupling are investigated, by solving the corresponding Gross–Pitaevskii equations, and via the time-splitting Fourier pseudospectral method. Taking into account the ferromagnetic interaction as well as the antiferromagnetic case, we identify some novel structures on the ground states, including the spot phase, particle-like supersolid phase, hole-like supersolid phase, particle-hole-like supersolid phase, and wave-shaped stripe phase. With the variance of helicoidal gauge potential, the system would undergo abundant quantum phases.

A general class of linear unconditionally energy stable schemes for the gradient flows, II

This paper continues to study linear and unconditionally modified-energy stable (abbreviated as SAV-GL) schemes for the gradient flows. The schemes are built on the SAV technique and the general linear time discretizations (GLTD) as well as the extrapolation for the nonlinear term. Different from [44], the GLTDs with three parameters discussed here are not necessarily algebraically stable. Some algebraic identities are derived by using the method of undetermined coefficients and further used to establish the modified-energy inequalities for the unconditional modified-energy stability of the semi-discrete-in-time SAV-GL schemes. It is worth emphasizing that those algebraic identities or energy inequalities are not necessarily unique for some choices of three parameters in the GLTDs. Numerical experiments on the Allen-Cahn, the Cahn-Hilliard and the phase field crystal models with the periodic boundary conditions are conducted to validate the unconditional modified-energy stability of the SAV-GL schemes, where the Fourier pseudo-spectral method is employed in space with the zero-padding to eliminate the aliasing error and the time stepsizes for ensuring the original-energy monotonous decay are estimated by using the stability regions of our SAV-GL schemes for the test equation. The resulting time stepsize constraints for the SAV-GL schemes are almost consistent with the numerical results on the above gradient flow models.

Energy-Preserving AVF Methods for Riesz Space-Fractional Nonlinear KGZ and KGS Equations

The Riesz space-fractional derivative is discretized by the Fourier pseudo-spectral (FPS) method. The Riesz space-fractional nonlinear Klein–Gordon–Zakharov (KGZ) and Klein–Gordon–Schrödinger (KGS) equations are transformed into two infinite-dimensional Hamiltonian systems, which are discretized by the FPS method. Two finite-dimensional Hamiltonian systems are thus obtained and solved by the second-order average vector field (AVF) method. The energy conservation property of these new discrete schemes of the fractional KGZ and KGS equations is proven. These schemes are applied to simulate the evolution of two fractional differential equations. Numerical results show that these schemes can simulate the evolution of these fractional differential equations well and maintain the energy-preserving property.

A New Framework of Convergence Analysis for Solving the General Nonlinear Schrödinger Equation Using the Fourier Pseudo-Spectral Method in Two Dimensions

A New Framework of Convergence Analysis for Solving the General Nonlinear Schrödinger Equation Using the Fourier Pseudo-Spectral Method in Two Dimensions

Improved uniform error bounds on time-splitting methods for the long-time dynamics of the weakly nonlinear Dirac equation

Abstract Improved uniform error bounds on time-splitting methods are rigorously proven for the long-time dynamics of the weakly nonlinear Dirac equation (NLDE), where the nonlinearity strength is characterized by a dimensionless parameter $\varepsilon \in (0, 1]$. We adopt a second-order Strang splitting method to discretize the NLDE in time, and combine with the Fourier pseudospectral method in space for the full-discretization. By employing the regularity compensation oscillation (RCO) technique, where the high frequency modes are controlled by the regularity of the exact solution, and the low frequency modes are analyzed by phase cancellation and energy method, we establish improved uniform error bounds at $O(\varepsilon ^{2}\tau ^{2})$ and $O(h^{m-1}+ \varepsilon ^{2}\tau ^{2})$ for the second-order Strang splitting semidiscretization and full-discretization up to the long-time $T_{\varepsilon } = T/\varepsilon ^{2}$ with $T>0$ fixed, respectively. Furthermore, the numerical scheme and error estimates are extended to an oscillatory NLDE, which propagates waves with $O(\varepsilon ^{2})$ wavelength in time and at $O(\varepsilon ^{-2})$ wave speed in space. Finally, numerical examples verifying our analytical results are given.

A uniformly accurate method for the Klein-Gordon-Dirac system in the nonrelativistic regime

A multiscale time integrator Fourier pseudospectral (MTI-FP) method is presented for discretizing the massive Klein-Gordon-Dirac (KGD) system which involves a small dimensionless parameter 0<ε≤1. In the nonrelativistic limit regime, the KGD system admits rapid oscillations in time as ε→0+. In addition, the nonlinear Yukawa interaction and the indefinite Dirac operator bring other significant difficulties. The main idea of the MTI-FP method is to construct a precise multiscale decomposition by the frequency (MDF) to the solution of the KGD system at each time step and then employ the Fourier pseudospectral discretization for the spatial derivatives followed with the exponential wave integrator (EWI) for the time marching. This approach is explicit, easy to implement and preforms significantly better than the classical methods in the literature. More specifically, we rigorously establish the uniform error bounds at O(τ+hm0−1) for all ε∈(0,1] and optimal quadratic temporal error bounds at O(τ2) in the ε=O(1) regime, where τ is the time step size, h is mesh size and m0 depends on the regularity of the solution. Extensive numerical results demonstrate that our error bounds are optimal and sharp. Finally, we apply the MTI-FP method to numerically study the nonrelativistic limit behaviors of the KGD system when ε→0+.

Two efficient exponential energy-preserving schemes for the fractional Klein–Gordon Schrödinger equation

This paper constructs two efficient exponential energy-preserving schemes for solving the fractional Klein–Gordon Schrödinger equation. The developed schemes are built upon the newly proposed partitioned averaged vector field method and exponential time difference technique and enjoy some advantages of the partitioned averaged vector field method. In addition, the Fourier pseudo-spectral method is applied to discretize the fractional Laplacian operator to obtain schemes so that the FFT technique can be used to reduce the computational complexity of the developed schemes in long-time simulations. Finally, by solving some fractional Klein–Gordon Schrödinger equations, it is demonstrated that the proposed schemes are efficient, conserve energy, and have better numerical stability results than traditional schemes.

Conformal structure‐preserving method for two‐dimensional damped nonlinear fractional Schrödinger equation

AbstractThis paper mainly analyzes conservation laws and convergence of splitting conformal multisymplectic scheme for solving the two‐dimensional damped nonlinear fractional Schrödinger equation. In order to obtain conformal multisymplectic scheme, first using Strang splitting method, the original problem is split into conservative multisymplectic and dissipative multisymplectic systems. The conservative multisymplectic system is numerically solved and the dissipative part is solved exactly. And then there shows the corresponding conservation laws and numerical scheme, in which the implicit midpoint method is used in time and the Fourier pseudospectral method is used in space, it also preserves conformal multisymplectic, the global conformal symplectic and global conformal mass conservation laws. Most important of all, we discuss convergence of the proposed scheme which is second‐order accuracy in time and spectral accuracy in space. Finally, the validity and accuracy of the theoretical results are verified by several numerical examples.

Flow Modeling over Airfoils and Vertical Axis Wind Turbines Using Fourier Pseudo-Spectral Method and Coupled Immersed Boundary Method

In the present work, verifying the applicability and potentiality of the IMERSPEC methodology for numerical and computational modeling of two-dimensional flows over airfoils and vertical axis wind turbines is proposed. It is a high-order convergence methodology with low computational cost when compared to other high-order methods, resulting from the coupling of the Fourier pseudo-spectral method and the immersed boundary method. To validate the proposed methodology, flow simulations are carried out over an airfoil NACA 0012 for a Reynolds number equal to 1000. From the spatial discretization procedure, there is convergence and good agreement of the lift and drag coefficients and the Strouhal number in relation to reference works. The behavior of the flows over the airfoil, as a function of the angle of attack, is evaluated by pressure and vorticity fields. From the analyzed flows, it is observed that the formation of different wake modes, constituted by swirling structures that vary their characteristic sizes, is influenced by the angle of attack. A case study is proposed based on the analysis of the main fluid dynamic aspects of flows over wind turbines with a vertical axis of three blades for a Reynolds number equal to 100. For this, a mathematical model responsible for the imposition of the rotational movement on the blades is presented in the turbine. Performance parameters, such as the coefficient of tangential force and normal force, and the analysis of velocity fields on the simulated turbine were presented and compared with other numerical methods. The good level of convergence and the accuracy of the obtained results show the promising capacity of the IMERSPEC methodology in solving problems of this nature.

Numerical Solutions of the (2+1)-Dimensional Nonlinear and Linear Time-Dependent Schrödinger Equations Using Three Efficient Approximate Schemes

In this paper, the (2+1)-dimensional nonlinear Schrödinger equation (2D NLSE) abreast of the (2+1)-dimensional linear time-dependent Schrödinger equation (2D TDSE) are thoroughly investigated. For the first time, these two notable 2D equations are attempted to be solved using three compelling pseudo-spectral/finite difference approaches, namely the split-step Fourier transform (SSFT), Fourier pseudo-spectral method (FPSM), and the hopscotch method (HSM). A bright 1-soliton solution is considered for the 2D NLSE, whereas a Gaussian wave solution is determined for the 2D TDSE. Although the analytical solutions of these partial differential equations can sometimes be reached, they are either limited to a specific set of initial conditions or even perplexing to find. Therefore, our suggested approximate solutions are of tremendous significance, not only for our proposed equations, but also to apply to other equations. Finally, systematic comparisons of the three suggested approaches are conducted to corroborate the accuracy and reliability of these numerical techniques. In addition, each scheme’s error and convergence analysis is numerically exhibited. Based on the MATLAB findings, the novelty of this work is that the SSFT has proven to be an invaluable tool for the presented 2D simulations from the speed, accuracy, and convergence perspectives, especially when compared to the other suggested schemes.