Pseudospectral Scheme Research Articles

Multi-Symplectic Method for the Two-Component Camassa–Holm (2CH) System

In this paper, the multi-symplectic formulations of the two-component Camassa–Holm system are presented. Both the multi-symplectic structure and two local conservation laws of the generalized two-component Camassa–Holm model are proposed for its first-order canonical form. Then, combining the Fourier pseudo-spectral method in the spatial domain with the midpoint method in the time dimension, the multi-symplectic Fourier pseudo-spectral scheme is constructed for the first-order canonical form. Meanwhile, the discrete scheme of the residuals of the multi-symplectic structure and two local conservation laws are also provided. By using the multi-symplectic Fourier pseudo-spectral scheme, the evolution of one- and two-soliton solutions for the generalized two-component Camassa–Holm model is regained. The structure-preserving properties and the reliability of the numerical scheme are illustrated by the tiny numerical residuals (less than 3.5 × 10−8) of the conservation laws as well as the tiny numerical variations (less than 1 × 10−9) of the amplitudes and the propagating velocities of the solitons.

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.

High-order $$L^{2}$$-bound-preserving Fourier pseudo-spectral schemes for the Allen-Cahn equation

High-order $$L^{2}$$-bound-preserving Fourier pseudo-spectral schemes for the Allen-Cahn equation

COMPOSITE LAGUERRE PSEUDOSPECTRAL METHOD FOR FOKKER-PLANCK EQUATIONS

A composite generalized Laguerre pseudospectral method for the nonlinear Fokker-Planck equations on the whole line is developed. Some composite generalized Laguerre interpolation approximation results are established. As an application, a composite Laguerre pseudospectral scheme is provided for the problems of the relaxation of fermion and boson gases. Convergence and stability of the scheme are proved. Numerical results show the efficiency of this approach and coincide well with theoretical analysis.

Direct–Indirect Hybrid Strategy for Optimal Powered Descent and Landing

A hybrid algorithm to solve optimal control problems is discussed in the present paper and applied to the powered descent guidance (PDG) problem. A reference solution is first obtained via a convex direct solver and is then used as guess for the primal–dual boundary-value problem associated with the initial problem. In this context, a covector mapping theorem is used to map the multipliers of the direct solution to the corresponding discrete costates of the indirect method. Collocation based on hp pseudospectral scheme is employed for the convex direct step, while single shooting is used for the indirect step. A switching-detection technique further equips the shooting. As opposed to the hybrid convex-indirect algorithm, a state-of-the-art purely indirect algorithm is outlined; such approach merges the same single shooting approach with a homotopic continuation. The proposed methods are applied to the pinpoint landing formulation of the PDG, framed in a three-dimensional environment. The results are finally outlined, comparing the proposed hybrid strategy to the purely indirect approach. The outcome highlights the gain in computational times for the hybrid optimization technique over the fully homotopic scheme, demonstrating the validity of the former for landing trajectory optimization purposes.

A fixed-point iteration method for high frequency vector wave equations

For numerically solving the high frequency vector wave equations, we present a simple approach based on fixed point iterations, where the problem is transferred into a fixed-point problem related to an exponential operator. The associated functional evaluations are achieved by unconditionally stable operator-splitting based pseudospectral schemes such that large step sizes are allowed to reach the approximated fixed point efficiently for prescribed termination criteria. For the sub-operator that is related to non-constant relative permeability, the Krylov subspace method or Taylor expansion is adapted for approximating its exponential. Furthermore, the Anderson acceleration is incorporated to accelerate the convergence of the fixed-point iterations. Numerical experiments are presented to demonstrate the accuracy and efficiency of the method.

A pseudospectral implicit particle-in-cell method with exact energy and charge conservation

The standard particle-in-cell (PIC) method employs explicit finite-difference (FD) schemes, such as the leap-frog scheme, for spatial and temporal integrations. In this study, we introduce a novel approach by employing a pseudospectral method for solving the Poisson equation and adopting a fully implicit time integration technique to achieve exact energy conservation. The pseudospectral field solver offers the advantage of spectral accuracy in obtaining field solutions. Previous investigations on implicit time integration of PIC FD equations have demonstrated the capability of enforcing precise energy exchange between the field and particles, resulting in energy-conserving schemes. We prove that the exact energy conservation property can be extended to the pseudospectral scheme. Additionally, we provide a solution to ensure a pseudospectral charge continuity equation. To validate our methodology, we implement the new scheme in a 2D electrostatic PIC code and present numerical examples that confirm the theoretical findings.

Improved finite-difference and pseudospectral schemes for the Kardar–Parisi–Zhang equation with long-range temporal correlations

Improved finite-difference and pseudospectral schemes for the Kardar–Parisi–Zhang equation with long-range temporal correlations

Improved uniform error bounds of a time-splitting Fourier pseudo-spectral scheme for the Klein–Gordon–Schrödinger equation with the small coupling constant

Recently, the long time numerical simulation of PDEs with weak nonlinearity (or small potentials) becomes an interesting topic. In this paper, for the Klein–Gordon–Schrödinger equation (KGSE) with a small coupling constant ɛ∈(0,1], we proposed a time-splitting Fourier pseudo-spectral (TSFP) scheme by reformulating the KGSE into a coupled nonlinear Schrödinger system (CNLSS). Through rigorous error analysis, we establish improved error bounds for the scheme at O(hm+ɛτ2) up to the long time at O(1/ɛ) where h is the mesh size and τ is the time step, respectively, and m depends on the regularity conditions. Compared with the results of existing numerical analysis, our analysis has the advantage of showing the long time numerical errors for the KGSE with the small coupling constant. The tools for error analysis mainly include the mathematical induction and the standard energy method as well as the regularity compensation oscillation (RCO) technique which has been developed recently. The numerical experiments support our theoretical analysis. Our scheme is novel because that to the best of our knowledge there has not been any TSFP scheme and any relevant long time analysis for the KGSE.

Optimal error estimates of a time-splitting Fourier pseudo-spectral scheme for the Klein–Gordon–Dirac equation

Recently, a time-splitting Fourier pseudo-spectral (TSFP) scheme for solving numerically the Klein–Gordon–Dirac equation (KGDE) has been proposed (Yi et al., 2019). However, that paper only gives numerical experiments and lacks rigorous convergence analysis and error estimates for the scheme. In addition, the time symmetry of the scheme has not been proved. This is not satisfactory from the perspective of geometric numerical integration. In this paper, we proposed a new TSFP scheme for the KGDE with periodic boundary conditions by reformulating the Klein–Gordon part into a relativistic nonlinear Schrödinger equation. The new scheme is time symmetric, fully explicit and conserves the discrete mass exactly. We make a rigorously convergence analysis and establish error estimates by comparing semi-discretization and full-discretization using the mathematical induction. The convergence rate of the scheme is proved to be second-order in time and spectral-order in space, respectively, in a generic norm under the specific regularity conditions. The numerical experiments support our theoretical analysis. The conclusion is also applicable to high-dimensional problems under sufficient regular conditions. Our scheme can also serve as a reference for solving some other coupled equations or systems such as Klein–Gordon–Schrödinger equation.

A conservative exponential integrators method for fractional conservative differential equations

<abstract><p>The paper constructs a conservative Fourier pseudo-spectral scheme for some conservative fractional partial differential equations. The scheme is obtained by using the exponential time difference averaged vector field method to approximate the time direction and applying the Fourier pseudo-spectral method to discretize the fractional Laplacian operator so that the FFT technique can be used to reduce the computational complexity in long-time simulations. In addition, the developed scheme can be applied to solve fractional Hamiltonian differential equations because the scheme constructed is built upon the general Hamiltonian form of the equations. The conservation and accuracy of the scheme are demonstrated by solving the fractional Schrödinger equation.</p></abstract>

Convergence analysis of two conservative finite difference fourier pseudo-spectral schemes for klein-Gordon-Dirac system

In this paper, we propose and study two conservative finite difference Fourier pseudo-spectral schemes numerically solving the Klein-Gordon-Dirac (KGD) system with periodic boundary conditions. The resulting numerical schemes are time symmetric and proved to conserve the discrete mass and the discrete energy. We give a rigorously convergence analysis for the schemes. Specifically, we establish the error estimates which are without any restrictions (CFL condition) on the ratio of time step to space step. The convergence rates of the new schemes are proved to be the temporal second-order and spatial spectral-order, respectively, in a Hm-norm. The main proof tools include the ideas of standard mathematical induction and the method of defining energy. Finally, we give the numerical experiments to support our theoretical analysis and error bounds.

Numerical approximations for the nonlinear time fractional reaction–diffusion equation

AbstractIn this article, we propose an implicit pseudospectral scheme for nonlinear time fractional reaction–diffusion equations with Neumann boundary conditions, which is based upon Gauss–Lobatto–Legendre–Birkhoff pseudospectral method in space and finite difference method in time. A priori estimate of numerical solution is given firstly. Then the existence of numerical solution is proved by Brouwer fixed point theorem and the uniqueness is obtained. It is proved rigorously that the fully discrete scheme is unconditionally stable and convergent. Furthermore, we develop a modified scheme by adding correction terms for the problem with nonsmooth solutions. Numerical examples are given to verify the theoretical analysis.

On the accuracy and spatial sampling of finite-difference modelling in discontinuous models

Finite-difference modelling estimates the wavefield in the subsurface by solving the elastic or acoustic wave equation numerically in a discrete version of the subsurface. The derivatives in the wave equation are approximated by their finite-difference counterparts. In this paper, we investigate the accuracy of acoustic finite-difference modelling as a function of the spatial sampling rate, the frequency, the source-receiver offset and the model parameters and parametrisation. To represent discontinuities in a regular grid, we apply a tapered low-pass wavenumber filter to densely sampled compliance and density models using the Nyquist wavenumber of the desired grid as cut-off wavenumber. A long filter taper improves the finite-difference modelling accuracy but also leads to longer oscillations in the model grids. We use a 2D acoustic central-grid pseudospectral scheme and compare it to analytical solutions of the wave equation and the reflectivity method. We show that previous recommendations for the spatial sampling of four grid points per shortest wavelength in staggered-grid schemes also apply to central-grid schemes. In the case of a single thin layer, the accuracy of finite-difference modelling is dependent on the impedance contrast and the layer thickness. For accurate wide-angle reflection amplitudes of a thin layer with a strong impedance contrast, a denser grid sampling than four grid points per shortest wavelength is required. Furthermore, we demonstrate that the presented wavenumber filtering approach is better suited for the downsampling of regularly sampled compliance and density data derived from well logs than Backus averaging.

An iterative framework to solve nonlinear optimal control with proportional delay using successive convexification and symplectic multi-interval pseudospectral scheme

In this paper, we propose an iterative framework to solve optimal control for nonlinear proportional state-delay systems. The successive convexification technique is first implemented to convert the original nonlinear problem into a sequence of linear-quadratic problems. And a symplectic pseudospectral method, where the multi-interval pseudospectral scheme is applied with a proportional mesh, to solve the transformed problems is then developed based on the first-order necessary conditions. Each linear-quadratic problem is finally transformed into a system of linear algebraic equations with a sparse coefficient matrix. Due to the benefit of the successive convexification technique and the multi-interval pseudospectral method, initial guess on costate variables is avoided and converged solutions can be obtained with an exponential convergent rate. The proposed iterative framework is validated by four examples with distinct features, highlighting its numerical precision and efficiency. And either exponential or linear convergence property can be exhibited by tuning the approximation degree or the mesh number.

Optimal guidance method based on conjugate gradient method and pseudospectral method

This paper aims to propose a method to solve the nonlinear optimal guidance problems with various performance indexes. First, the conjugate gradient method is derived based on the performance indexes of general form and specified form, and the execution steps of the optimal guidance method are obtained. Secondly, the pseudospectral collocation scheme is used to solve some intermediate variables in the algorithm, and the differential equations are transformed into algebraic equations, so as to improve the computational efficiency. Finally, the proposed method is applied to the air-to-ground missile guidance problem. By comparing with the simulation results of GPOPS-II, the optimality of the proposed method is verified. Additionally, the high computational efficiency of the algorithm is also verified.

Kinetic roughening in the nonlocal Kardar–Parisi–Zhang growth: Pseudospectral versus finite difference schemes

To investigate the effects of nonlocal interaction in kinetic roughening, we adopt two independent numerical methods including pseudospectral (PS) and finite difference (FD) schemes to simulate the nonlocal Kardar–Parisi–Zhang (NKPZ) equation in (1+1)-dimensions. We find that the values of growth exponent β increase gradually with the spatial decay exponent ρ, indicating that the scaling properties are dependence on nonlocal interactions within the effective spatial decay region. However, our simulations also imply that both the local roughness exponent αloc and the spectral roughness exponent αs decrease slightly with ρ. Our results could be consistent with each other using these two numerical methods, and the comparisons and differences among the existing analytical predictions and numerical results are also discussed.

Pell Collocation Pseudo Spectral Scheme for One-Dimensional Time-Fractional Convection Equation with Error Analysis

In the current manuscript, we implement the collocation method to obtain an approximate solution of one-dimensional time-fractional convection equation. The operational matrices of Pell polynomials are applied to solve the fractional partial differential equations. In the Caputo sense, we describe the time-fractional derivatives. So, this algorithm allows us to transform the fractional differential equation with initial (boundary) conditions into a system of algebraic equations in a good form. The convergence and error analysis of Pell polynomials are carefully investigated, and some examples are explored to show the comparison and efficiency of our method with others.

Preconditioned Pseudo-Spectral Gradient Flow for Computing the Steady-State of Space Fractional Cahn-Allen Equations With Variable Coefficients

The aim of this paper is to propose some efficient and accurate numerical methods to compute the steady-state of variable coefficients space fractional Cahn-Allen equations. The approach combines an adaptive time stepping semi-implicit gradient flow method to minimize the fractional energy functional and pseudo-spectral approximation schemes. Based on the use of a preconditioned GMRES, the space fractional Cahn-Allen equation is then solved efficiently. The full methodology is supported by the numerical solution of a one-dimensional problem.

A pseudo-spectral based efficient volume penalization scheme for Cahn–Hilliard equation in complex geometries

In this paper, we have developed an efficient volume penalization based diffuse-filtering scheme to solve variable mobility based Cahn–Hilliard equation in complex geometries. The main novelty of the work is that a dealiased pseudo-spectral scheme with immersed interface method (IIM) is proposed for solving any generalized concentration-dependent mobility function-based Cahn–Hilliard (CH) equation in complicated computational domains. An indicator function based mobility parameter is introduced to perform simulation of binary spinodal decomposition problem at a lower computational expense in complex geometries by solving a single phase field equation. Due to the smooth removal of high frequency Fourier components, the solution of the present RK4 based diffuse-filtering scheme does not display spurious currents when suitable low-pass filtering strategy and adequately resolved mobility indicator are incorporated. The traditional and memory optimized zero padding schemes are also implemented to show the comparative performance of different dealiasing schemes for the variable mobility based Cahn–Hilliard equation. It is found that the diffuse-filtering scheme displays reasonable accuracy similar to the zero padding based schemes but its average CPU time is significantly lower, which indicates better computational performance of the scheme for the variable mobility Cahn–Hilliard equation. Time variation of the characteristic length scale during spinodal decomposition of a binary mixture agrees well with the analytical prediction. The optimal three stage SSPRK3 temporal scheme is employed and it is found that time step size can be increased approximately 1.4 times than the classical RK4 scheme reducing total CPU time. Oscillation free numerical solution and conservation of order parameter are obtained for the complex geometry based spinodal decomposition problem. A radially-averaged structure factor is introduced to quantify resolution issues of the dealiasing schemes for the spinodal decomposition problem in different complex geometries.