Fourier Spectral Method Research Articles

Kernel Determination Problem in the Third Order 1D Moore-Gibson-Thompson Equation with Memory

In this study, we address the inverse problem of determining the convolution kernel function in the third-order Moore--Gibson--Thompson (MGT) equation, which is commonly used to model fluid motion with memory effects. Specifically, we focus on the determination of the unknown kernel, which governs the memory term in the equation. Initially, we employ the Fourier spectral method to solve the direct initial-boundary-value problem for a non-homogeneous MGT equation that with the memory term. The Fourier spectral method allows us to leverage the problem's inherent linearity and spatial homogeneity, leading to an efficient and explicit construction of the solution. The direct problem is analyzed under appropriate initial and boundary conditions, which are carefully specified to ensure mathematical consistency. To solve the inverse problem, we introduce an additional condition-typically a form of observational data such as at certain points-which provides the necessary constraints for determining the kernel. We prove local existence and uniqueness theorems for solution of the problem.

A least-squares Fourier frame method for nonlocal diffusion models on arbitrary domains

We introduce a least-squares Fourier frame method for solving nonlocal diffusion models with Dirichlet volume constraint on arbitrary domains. The mathematical structure of a frame rather than a basis allows using a discrete least-squares approximation on irregular domains and imposing non-periodic boundary conditions. The method has inherited the one-dimensional integral expression of Fourier symbols of the nonlocal diffusion operator from Fourier spectral methods for any d spatial dimensions. High precision of its solution can be achieved via a direct solver such as pivoted QR decomposition even though the corresponding system is extremely ill-conditioned, due to the redundancy in the frame. The extension of AZ algorithm improves the complexity of solving the rectangular linear system to O(Nlog2⁡N) for 1d problems and O(N2log2⁡N) for 2d problems, compared with O(N3) of the direct solvers, where N is the number of degrees of freedom. We present ample numerical experiments to show the flexibility, fast convergence and asymptotical compatibility of the proposed method.

The localized radial basis function collocation method for dendritic solidification, solid phase sintering and wetting phenomenon based on phase field

Phase-field has been effectively applied to many complex problems according to the mesh based method. However, the computational speed of the numerical method based on phase-field still needs improved. In this paper, an improved localized radial basis function collocation method (LRBFCM) based on the adaptive support domain is employed to the phase-field methods. The proposed adaptive support domain can increase the stability of the LRBFCM, and the improved LRBFCM is much more efficient than the traditional finite element method (FEM) in coupling with phase-field methods. The proposed approach is further applied to the single-phase dendrite solidification, two-phase sintering, and three-phase wetting phenomena. We compare the efficiency of the proposed LRBFCM with different numerical methods, which show that the LRBFCM combined with the Fourier spectral method can deal with the three-phase model with more than ten million nodes easily.

Evolution of water wave packets by wind in shallow water

We use the Korteweg–de Vries (KdV) equation, supplemented with several forcing/friction terms, to describe the evolution of wind-driven water wave packets in shallow water. The forcing/friction terms describe wind-wave growth due to critical level instability in the air, wave decay due to laminar friction in the water at the air–water interface, wave stress in the air near the interface induced by a turbulent wind and wave decay due to a turbulent bottom boundary layer. The outcome is a modified KdV–Burgers equation that can be a stable or unstable model depending on the forcing/friction parameters. To analyse the evolution of water wave packets, we adapt the Whitham modulation theory for a slowly varying periodic wave train with an emphasis on the solitary wave train limit. The main outcome is the predicted growth and decay rates due to the forcing/friction terms. Numerical simulations using a Fourier spectral method are performed to validate the theory for various cases of initial wave amplitudes and growth and/or decay parameter ranges. The results from the modulation theory agree well with these simulations. In most cases we examined, many solitary waves are generated, suggesting the formation of a soliton gas.

An unconditionally energy dissipative, adaptive IMEX BDF2 scheme and its error estimates for Cahn–Hilliard equation on generalized SAV approach

Abstract An adaptive implicit-explicit (IMEX) BDF2 scheme is investigated on generalized SAV approach for the Cahn–Hilliard equation by combining with Fourier spectral method in space. It is proved that the modified energy dissipation law is unconditionally preserved at discrete levels. Under a mild ratio restriction, i.e., A1: $0<r_{k}:=\tau _{k}/\tau _{k-1}< r_{\max }\approx 4.8645$, we establish a rigorous error estimate in $H^{1}$-norm and achieve optimal second-order accuracy in time. The proof involves the tools of discrete orthogonal convolution (DOC) kernels and inequality zoom. It is worth noting that the presented adaptive time-step scheme only requires solving one linear system with constant coefficients at each time step. In our analysis, the first-consistent BDF1 for the first step does not bring the order reduction in $H^{1}$-norm. The $H^{1}$ bound of numerical solution under periodic boundary conditions can be derived without any restriction (such as zero mean of the initial data). Finally, numerical examples are provided to verify our theoretical analysis and the algorithm efficiency.

An exponential spectral deferred correction method for multidimensional parabolic problems

We present some efficient algorithms based on an exponential time differencing spectral deferred correction (ETDSDC) method for multidimensional second and fourth-order parabolic problems with non-periodic boundary conditions including Dirichlet, Neumann, Robin boundary conditions. Similar to the Fourier spectral method for periodic problems, the key to the efficiency of our algorithms is to construct diagonal discrete linear operators via Legendre–Galerkin methods with Fourier-like basis functions. In combination with the ETDSDC scheme, the proposed methods are spectrally accurate in space and up to 10th-order accurate in time (as shown in this work). We demonstrate the high-order of convergence and efficiency of our algorithms in solving parabolic equations through a series of two-dimensional and three-dimensional examples including Ginzburg–Landau and Allen–Cahn equations.

Coarse-Gridded Simulation of the Nonlinear Schrödinger Equation with Machine Learning

A numerical method for evolving the nonlinear Schrödinger equation on a coarse spatial grid is developed. This trains a neural network to generate the optimal stencil weights to discretize the second derivative of solutions to the nonlinear Schrödinger equation. The neural network is embedded in a symmetric matrix to control the scheme’s eigenvalues, ensuring stability. The machine-learned method can outperform both its parent finite difference method and a Fourier spectral method. The trained scheme has the same asymptotic operation cost as its parent finite difference method after training. Unlike traditional methods, the performance depends on how close the initial data are to the training set.

Direct simulation of vortex dynamics of multi-cellular Taylor–Green vortex by pseudo-spectral method

This study investigates the long-time vorticity dynamics of the multi-cellular configurations of the two-dimensional (2D) Taylor–Green vortex (TGV). The pseudo-spectral method is used to solve the incompressible Navier–Stokes equation to analyze the evolution of TGV arrays. The focus is on understanding vortex interactions leading to vortex filamentation and stripping (forward cascade) during primary instability; merger and reconnection (inverse cascade) among the TGV vortical cells subsequently. Here, consideration of multiple cells avoids imposing symmetries at the smallest periodic length scale, and thereby affecting disturbance growth. The initial condition is taken from the analytic solution of the TGV, and Fourier spectral method is employed to track the interactions of the initial doubly-periodic vortices. The full sequence of evolution from one equilibrium state to another for the TGV is not addressed before, as reported here to fill this gap for multiple TGV cells in both directions. By studying various vortical interactions in the ensemble, here we report the enstrophy and energy spectra for different number of TGV cells. This is crucial in understanding the very long-time evolution process, at post-critical Reynolds numbers for the 2D TGV problem in the same physical domain, (0≤(x,y)≤4π) having (4×4) and (6×6) cells. Reported results show the evolution of these vortical cells from original configurations to finally a (1×1) vortical cells — the universal state not demonstrated before.

Fast implicit update schemes for Cahn–Hilliard-type gradient flow in the context of Fourier-spectral methods

This work discusses a way of allowing fast implicit update schemes for the temporal discretization of phase-field models for gradient flow problems that employ Fourier-spectral methods for their spatial discretization. Through the repeated application of the Sherman–Morrison formula we provide a rule for approximations of the inverted tangent matrix of the corresponding Newton–Raphson method with a selectable order. Since the representation of this inversion is exact for a sufficiently high approximation order, the proposed scheme is shown to provide a fixed-point-type iterative solver for gradient flow problems that require the solution of linear systems in the context of an implicit time-integration. While the proposed scheme is applicable to general gradient flow phase-field models, we discuss the scheme in the context of the Cahn–Hilliard equation, the Swift–Hohenberg equation, and the phase-field crystal equation for which we demonstrate the performance of the proposed method in comparison with classical solvers.

Asymptotically compatible schemes for nonlocal Ohta–Kawasaki model

AbstractWe study the asymptotical compatibility of the Fourier spectral method in multidimensional space for the nonlocal Ohta–Kawasaka model, a more generalized version of the model proposed in our previous work (Y. Zhao and W. Luo, Physica D 458 (2024), 133989). By introducing the Fourier collocation discretization for the spatial variable, we show that the asymptotical compatibility holds in 2D and 3D over a periodic domain. For the temporal discretization, we adopt the second‐order backward differentiation formula method. We prove that for certain nonlocal kernels, the proposed time discretization schemes inherit the energy dissipation law. In the numerical experiments, we verify the asymptotical compatibility, the second‐order temporal convergence rate, and the energy stability of the proposed schemes. More importantly, we discover a novel square lattice pattern when certain nonlocal kernel are applied in the model. In addition, our numerical experiments confirm the existence of an upper bound for the optimal number of bubbles in 2D for some specific nonlocal kernels. Finally, we numerically explore the promotion/demotion effect induced by the nonlocal horizon , which is consistent with the theoretical studies presented in our earlier work (Y. Zhao and W. Luo, Physica D 458 (2024), 133989).

On the convergence of Fourier spectral methods involving noncompact operators

Abstract Motivated by Fredholm theory, we develop a framework to establish the convergence of spectral methods for operator equations $\mathcal L u = f$. The framework posits the existence of a left-Fredholm regulator for $\mathcal L$ and the existence of a sufficiently good approximation of this regulator. Importantly, the numerical method itself need not make use of this extra approximant. We apply the framework to Fourier finite-section and collocation-based numerical methods for solving differential equations with periodic boundary conditions and to solving Riemann–Hilbert problems on the unit circle. We also obtain improved results concerning the approximation of eigenvalues of differential operators with periodic coefficients.

Dynamical Behavior of the Fractional BBMB Equation on Unbounded Domain

The fractional-order Benjamin-Bona-Mahony-Burgers (BBMB) equation is a generalization of the classical BBMB equation. It’s dynamic behaviors is much more complex than that of the corresponding integer-order BBMB equation. The main purpose of this paper is to explore the dynamic behaviors of the fractional-order BBMB equations by using the Fourier spectral method. Firstly, the numerical solution is compared with the exact solution. It is proved that the proposed method is effective and high precision for solving the spatial fractional order BBMB equation. Then, some dynamical behaviors of fractional order BBMB equations are obtained by using the present method, and some novel fractal waves of the the fractional-order BBMB equation on unbounded domain are shown.

Fourier spectral exponential time-differencing method for space-fractional generalized wave equations

Fourier spectral exponential time-differencing method for space-fractional generalized wave equations

An energy-stable variable-step L1 scheme for time-fractional Navier–Stokes equations

We propose a structure-preserving scheme and its error analysis for time-fractional Navier–Stokes equations (TFNSEs) with periodic boundary conditions. The equations are first rewritten as an equivalent system by eliminating the pressure explicitly. Then, the spatial and temporal discretization are done by the Fourier spectral method and variable-step L1 scheme, respectively. It is proved that the fully-discrete scheme is energy-stable and divergence-free. The energy is an asymptotically compatible one since it recovers the classical energy when α→1. Moreover, optimal error estimates are presented very technically by the obtained boundedness of the numerical solutions and some Sobolev inequalities. To our knowledge, they are the first results of the construction and analysis of structure-preserving schemes for TFNSEs. Several interesting numerical examples are given to confirm the theoretical results at last.

A second-order Strang splitting scheme for the generalized Allen–Cahn type phase-field crystal model with FCC ordering structure

In this paper, we consider the generalized Allen–Cahn-type phase-field crystal model with face-centered-cubic ordering structure (PFC-FCC). Due to the combined complexity of the eighth-order spatial derivative and inherent nonlinearity, it poses a significant challenge to design a numerical scheme of high accuracy, stability, and efficiency to solve the PFC-FCC model. Endeavoring towards this objective, we adopt operator splitting method to address the PFC-FCC model. Based on Fourier spectral method for spatial discretization and SSP-RK method for temporal discretization, we propose an effective and easy-to-implement second-order scheme. We are also able to proffer an analytical proof for discrete mass conservation, and engage a discussion on an optimal error estimate for the second-order scheme. In addition, the energy stability of the numerical scheme is derived. Ultimately, a series of numerical experiments specifically designed to substantiate its accuracy, efficiency, and capability for phase transition are performed.

On the phase field based model for the crystalline transition and nucleation within the Lagrange multiplier framework

Understanding the complexity of the nucleation and transition between the crystalline and quasicrystalline is significant because the structural incommensurability is anisotropic and of significance in revealing material properties. This paper reports the two- and three-dimensional nucleation and transition investigation from quasicrystals to crystals using a phase field method. The investigation starts with the Lifshitz–Petrich (LP) model, which is derived from the Landau theory for the exploration of critical nuclei. By the variational derivation, we construct two phase field models with tenth- and eighth-order, which provide the possibility of exploring the transition under stable phase states. In order to dissipate the original Lifshitz–Petrich energy, we apply a Lagrange multiplier method to modify the two models and solve them by the Fourier spectral method. Whereas the nonlinearity leads to expensive computational burden and extra stiffness, the designed algorithm can effectively avoid the numerical oscillations caused by rigidity and keep an O(Nlog⁡N) computational complexity, where N is the mesh grid size. To further demonstrate the robustness and advantages of the proposed method for handling phase-field modeling of crystalline structures, we compare its performance with other methods for constructing unconditionally stable methods. Our method can be directly implemented on a GPU for acceleration and achieves multiple times faster performance compared to CPU-only alternatives. Various numerical tests have been used to validate that our method works well for revealing the transition between different stable states during the nucleation process.

A Fourier spectral immersed boundary method with exact translation invariance, improved boundary resolution, and a divergence-free velocity field

This paper introduces a new immersed boundary (IB) method for viscous incompressible flow, based on a Fourier spectral method for the fluid solver and on the nonuniform fast Fourier transform (NUFFT) algorithm for coupling the fluid with the immersed boundary. The new Fourier spectral immersed boundary (FSIB) method gives improved boundary resolution in comparison to the standard IB method. The interpolated velocity field, in which the boundary moves, is analytically divergence-free. The FSIB method is gridless and has the meritorious properties of volume conservation, exact translation invariance, conservation of momentum, and conservation of energy. We verify these advantages of the FSIB method numerically both for the Stokes equations and for the Navier-Stokes equations in both two and three space dimensions. The FSIB method converges faster than the IB method. In particular, we observe second-order convergence in various problems for the Navier-Stokes equations in three dimensions. The FSIB method is also computationally efficient with complexity of O(N3log⁡(N)) per time step for N3 Fourier modes in three dimensions.

A-stable spectral deferred correction method for nonlinear Allen-Cahn model

This paper presents a type of A-stable spectral deferred correction (SDC) method. The scheme is initiated by the first-order backward Euler method. We adopt the linear stabilization approach for the Allen-Cahn model to get the linear semi-implicit SDC scheme. This is done by the addition and subtraction of the linear stabilization operators that have been provided for the Allen-Cahn problem. The ranges of the stabilization factors are then given. The aim is to achieve A-stability and error estimates for the implicit part of the scheme. Finally, the semi-implicit SDC scheme coupled with the Fourier spectral method is used to simulate the Allen-Cahn problem, which is also suitable for the highly nonlinear problem. Numerical experiments are given to numerically demonstrate the high-order accuracy and the energy decay property of the scheme with A-stability parameters.

A linearly implicit energy-stable scheme for critical dissipative surface quasi-geostrophic flows

In this paper, we propose an effective linearly implicit unconditional energy-stable scheme for surface quasi-geostrophic flows based on the scalar auxiliary variable approach and the Fourier spectral Galerkin method. Compared with traditional numerical methods, our scheme has constant coefficient matrices at each time step, and the numerical solutions are consistent with the dissipation laws for modified energy. By treating linear terms implicitly and nonlinear terms explicitly, we derive the dissipation laws for discrete modified surface kinetic energy and Hamiltonian. To reduce the aliasing error induced by the Fourier spectral Galerkin method, we implement a 2/3 de-aliasing technique for the nonlinear terms. Furthermore, the integration concerning energy in our numerical scheme is exact due to the Fourier spectral Galerkin method. Numerical experiments are presented to verify the stability and efficiency of the proposed scheme.

The Allen–Cahn model with a time-dependent parameter for motion by mean curvature up to the singularity

In this article, we investigate the temporal evolution of arbitrary, simple, closed two-dimensional (2D) and three-dimensional (3D) interfaces under motion driven by mean curvature up to a singularity. To facilitate this investigation, we propose a novel Allen–Cahn (AC) model with a time-dependent interfacial thickness parameter. The original AC equation was developed to model the phase separation of a binary mixture. It is well known that a level set or interface of the solution of the AC equation obeys the dynamics of motion by curvature as the interfacial thickness parameter approaches zero. Generally, it is difficult to find a closed-form analytic solution of the AC equation with any initial condition. Therefore, we need to estimate the solution of the AC equation through computational approaches such as finite difference method (FDM), finite element method (FEM), Fourier-spectral method (FSM), and finite volume method (FVM). Any simple, closed curves and convex surfaces eventually shrink to a point due to motion by mean curvature. Therefore, it becomes necessary to use adaptive mesh techniques to resolve this small size problem. However, even though we use adaptive mesh techniques, we still confront the relatively thick interfacial transition layer when the curves or interfaces become very small. To avoid this problem, we can start with a very small mesh size for a small value of the interfacial parameter, which results in an extremely high computational cost even when using adaptive mesh techniques. To resolve these issues, we present the AC equation with a time-dependent interfacial parameter and develop an adaptive mesh refinement system. To show the superior performance of the proposed mathematical equation and its computational algorithm, we present various numerical experiments and investigate the motion by mean curvature up to the singularity of curves in 2D space and interfaces in 3D space.