Fourier Spectral Method Research Articles

Single-distance phase retrieval algorithm for Bragg Magnifier microscope.

We present an improved, single-distance phase retrieval algorithm applicable for holographic X-ray imaging of biological objects for an in-line germanium Bragg Magnifier Microscope (BMM). The proposed algorithm takes advantage of a modified shrink-wrap algorithm for phase objects, robust unwrapping algorithm as well as other reasonable constraints applied to the wavefield at the object and the detector plane. The performance of the algorithm is analyzed on phantom objects and the results are shown and discussed. We demonstrated the suitability of the algorithm for the phase retrieval on a more complex biological specimen Tardigrade, where we achieved successful phase retrieval from only a single hologram. The spatial resolution obtained by Fourier spectral power method for biological objects is ∼ 300 nm, the same value as obtained from the reconstructed test pattern. Our results achieved using the new algorithm confirmed the potential of BMM for in-vivo, dose-efficient single-shot imaging of biological objects.

An efficient numerical method for the modified regularized long wave equation using Fourier spectral method

The modified regularized long wave (MRLW) equation is numerically solved using Fourier spectral collection method. The MRLW equation is discretized in space variable by the Fourier spectral method and Leap-Frog method for time dependence. To validate the efficiency, accuracy and simplicity of the used method, four cases study are solved. The single soliton wave motion, interaction of two solitary waves, interaction of three solitary waves and a Maxwellian initial condition pulse are studied. The L2 and L∞ error norms are computed for the motion of single solitary waves. To determine the conservation properties of the MRLW equation three invariants of motion are evaluated for all test problems.

Vibration analysis of coupled conical-cylindrical-spherical shells using a Fourier spectral element method.

This paper presents a Fourier spectral element method (FSEM) to analyze the free vibration of conical-cylindrical-spherical shells with arbitrary boundary conditions. Cylindrical-conical and cylindrical-spherical shells as special cases are also considered. In this method, each fundamental shell component (i.e., cylindrical, conical, and spherical shells) is divided into appropriate elements. The variational principle in conjunction with first-order shear deformation shell theory is employed to model the shell elements. Since the displacement and rotation components of each element are expressed as a linear superposition of nodeless Fourier sine functions and nodal Lagrangian polynomials, the global equations of the coupled shell structure can be obtained by adopting the assembly procedure. The Fourier sine series in the displacement field is introduced to enhance the accuracy and convergence of the solution. Numerical results show that the FSEM can be effectively applied to vibration analysis of the coupled shell structures. Numerous results for coupled shell structures with general boundary conditions are presented. Furthermore, the effects of geometric parameters and boundary conditions on the frequencies are investigated.

Spectral Domain Decomposition Using Local Fourier Basis: Application to Ultrasound Simulation on a Cluster of GPUs

The simulation of ultrasound wave propagation through biological tissue has a wide range of practical applications. However, large grid sizes are generally needed to capture the phenomena of interest. Here, a novel approach to reduce the computational complexity is presented. The model uses an accelerated k -space pseudospectral method which enables more than one hundred GPUs to be exploited to solve problems with more than 3*10^9 grid points. The classic communication bottleneck of Fourier spectral methods, all-to-all global data exchange, is overcome by the application of domain decomposition using local Fourier basis. Compared to global domain decomposition, for a grid size of 1536 x 1024 x 2048, this reduces the simulation time by a factor of 7.5 and the simulation cost by a factor of 3.8.

A Fourier spectral method for fractional-in-space Cahn–Hilliard equation

In this paper, a fractional extension of the Cahn–Hilliard (CH) phase field model is proposed, i.e. the fractional-in-space CH equation. The fractional order controls the thickness and the lifetime of the interface, which is typically diffusive in integer order case. An unconditionally energy stable Fourier spectral scheme is developed to solve the fractional equation with periodic or Neumann boundary conditions. This method is of spectral accuracy in space and of second-order accuracy in time. The main advantages of this method are that it yields high precision and high efficiency. Moreover, an extra stabilizing term is added to obey the energy decay property while maintaining accuracy and simplicity. Numerical experiments are presented to confirm the accuracy and effectiveness of the proposed method.

Mathematical analysis and numerical simulation of patterns in fractional and classical reaction-diffusion systems

The aim of this paper is to examine pattern formation in the sub— and super-diffusive scenarios and compare it with that of classical or standard diffusive processes in two-component fractional reaction-diffusion systems that modeled a predator-prey dynamics. The focus of the work concentrates on the use of two separate mathematical techniques, we formulate a Fourier spectral discretization method as an efficient alternative technique to solve fractional reaction-diffusion problems in higher-dimensional space, and later advance the resulting systems of ODEs in time with the adaptive exponential time-differencing solver. Obviously, the fractional Fourier approach is able to achieve spectral convergence up to machine precision regardless of the fractional order α, owing to the fact that our approach is able to give full diagonal representation of the fractional operator. The complexity of the dynamics in this system is theoretically discussed and graphically displayed with some examples and numerical simulations in one, two and three dimensions.

Morphomechanics of bacterial biofilms undergoing anisotropic differential growth

Growing bacterial biofilms exhibit a number of surface morphologies, e.g., concentric wrinkles, radial ridges, and labyrinthine networks, depending on their physiological status and nutrient access. We explore the mechanisms underlying the emergence of these greatly different morphologies. Ginzburg-Landau kinetic method and Fourier spectral method are integrated to simulate the morphological evolution of bacterial biofilms. It is shown that the morphological instability of biofilms is triggered by the stresses induced by anisotropic and heterogeneous bacterial expansion, and involves the competition between membrane energy and bending energy. Local interfacial delamination further enriches the morphologies of biofilms. Phase diagrams are established to reveal how the anisotropy and spatial heterogeneity of growth modulate the surface patterns. The mechanics of three-dimensional microbial morphogenesis may also underpin self-organization in other development systems and provide a potential strategy for engineering microscopic structures from bacterial aggregates.

Analysis of the operator splitting scheme for the Allen–Cahn equation

ABSTRACTThis paper presents two numerical schemes for solving the Allen–Cahn equation representing a model for antiphase domain coarsening in a binary mixture. Based on the operator splitting method, the Allen–Cahn equation was divided into a linear and a nonlinear sub equation: the linear sub equation was solved by a Fourier spectral method, which is based on the exact solution and thus has no stability restriction on the time-step size; the nonlinear sub equation was then solved analytically due to the availability of a closed-form solution. The stability and error analysis of the numerical solution are presented in detail. Numerical experiments are presented to confirm the accuracy, efficiency, and stability of the proposed method. In particular, we show that the schemes are unconditionally stable and second-order accurate in time.

An improved semi-Lagrangian time splitting spectral method for the semi-classical Schrödinger equation with vector potentials using NUFFT

In this paper, we propose a new time splitting Fourier spectral method for the semi-classical Schrödinger equation with vector potentials. Compared with the results in [21], our method achieves spectral accuracy in space by interpolating the Fourier series via the NonUniform Fast Fourier Transform (NUFFT) algorithm in the convection step. The NUFFT algorithm helps maintain high spatial accuracy of Fourier method, and at the same time improve the efficiency from O(N2) (of direct computation) to O(Nlog⁡N) operations, where N is the total number of grid points. The kinetic step and potential step are solved by analytical solution with pseudo-spectral approximation, and, therefore, we obtain spectral accuracy in space for the whole method. We prove that the method is unconditionally stable, and we show improved error estimates for both the wave function and physical observables, which agree with the results in [3] for vanishing potential cases and are superior to those in [21]. Extensive one and two dimensional numerical studies are presented to verify the properties of the proposed method, and simulations of 3D problems are demonstrated to show its potential for future practical applications.

Robust and adaptive techniques for numerical simulation of nonlinear partial differential equations of fractional order

In this paper, some nonlinear space-fractional order reaction-diffusion equations (SFORDE) on a finite but large spatial domain x ∈ [0, L], x=x(x,y,z) and t ∈ [0, T] are considered. Also in this work, the standard reaction-diffusion system with boundary conditions is generalized by replacing the second-order spatial derivatives with Riemann-Liouville space-fractional derivatives of order α, for 0 < α < 2. Fourier spectral method is introduced as a better alternative to existing low order schemes for the integration of fractional in space reaction-diffusion problems in conjunction with an adaptive exponential time differencing method, and solve a range of one-, two- and three-components SFORDE numerically to obtain patterns in one- and two-dimensions with a straight forward extension to three spatial dimensions in a sub-diffusive (0 < α < 1) and super-diffusive (1 < α < 2) scenarios. It is observed that computer simulations of SFORDE give enough evidence that pattern formation in fractional medium at certain parameter value is practically the same as in the standard reaction-diffusion case. With application to models in biology and physics, different spatiotemporal dynamics are observed and displayed.

Solution of stochastic nonlinear time fractional PDEs using polynomial chaos expansion combined with an exponential integrator

The aim of this paper is to introduce an efficient numerical algorithm for the solution of stochastic time fractional stiff partial differential equations (PDEs). The time fractional derivative is described in the Caputo sense. By applying polynomial chaos (PC) expansion to discretize the random variable, a coupled time fractional deterministic system of PDEs is obtained. For the resulting system of time fractional PDEs we apply the Fourier spectral method to discretize the spatial variable, and use the fast Fourier transform (FFT) during the computation. Then, we use an exponential integrator method to overcome the stability issues due to the stiffness in the resulting time fractional semi-discrete system. The considered models have two challenging parameters, fractional order of equations and noise amplitude. In addition to exponential integrator, we also implemented a predictor–corrector method of Adams–Bashforth–Moulton type. We include numerical results of applying the developed method to stochastic time fractional Burgers and Kuramoto–Sivashinsky (KS) equations, for various values of noise intensity. A comparison of performance of the proposed scheme with fractional Adams method is also reported which confirms the efficiency and applicability of the proposed method based on the exponential integrator scheme.

On Second Order Semi-implicit Fourier Spectral Methods for 2D Cahn–Hilliard Equations

We consider several seconder order in time stabilized semi-implicit Fourier spectral schemes for 2D Cahn---Hilliard equations. We introduce new stabilization techniques and prove unconditional energy stability for modified energy functionals. We also carry out a comparative study of several classical stabilization schemes and identify the corresponding stability regions. In several cases the energy stability is proved under relaxed constraints on the size of the time steps. We do not impose any Lipschitz assumption on the nonlinearity. The error analysis is obtained under almost optimal regularity assumptions.

Cross-diffusion-driven gravitational instability in a Hele-Shaw cell saturated with a ternary solution

The effect of cross diffusion on the onset of the gravitational instabilities in a Hele-Shaw cell saturated with a ternary solution is analyzed. Based on the linear stability theory, new stability equations are derived in the global domain and then transformed into the similar domain. These stability equations are solved by employing various methods such as an initial value problem approach and quasi-steady state approximations (QSSA’s). Through the initial growth rate analysis without the QSSA, we prove that initially the system is unconditionally stable. However, the QSSA in the global domain showed that the system can be initially unstable for a certain condition. Based on the QSSA in the similar domain (QSSAζ), we obtain the critical time for the onset of instability motion. As expected, the higher δ21β makes the system more unstable, i.e., accelerates the onset of instability motion; here δ21 and β represent the normalized cross diffusion coefficient and the ratio of densification coefficients, respectively. Based on the linear analysis, fully nonlinear analyses are also conducted by using the Fourier spectral method. The present nonlinear analyses show that the double-diffusive and diffusive-layer convection-type of instabilities are possible for the positive and negative δ21β-values, respectively. From the present nonlinear analysis, the system having δ22 &amp;gt; 1 prefers the instabilities with a larger wavelength than the system having δ22 &amp;lt; 1. Here δ22 is the normalized normal diffusion coefficient of component B.

Practical estimation of a splitting parameter for a spectral method for the ternary Cahn–Hilliard system with a logarithmic free energy

We propose a practical estimation of a splitting parameter for a spectral method for the ternary Cahn–Hilliard system with a logarithmic free energy. We use Eyre's convex splitting scheme for the time discretization and a Fourier spectral method for the space variables. Given an absolute temperature, we find composition values that make the total free energy be minimum. Then, we find the splitting parameter value that makes the two split homogeneous free energies be convex on the neighborhood of the local minimum concentrations. For general use, we also propose a sixth‐order polynomial approximation of the minimum concentration and derive a useful formula for the practical estimation of the splitting parameter in terms of the absolute temperature. The numerical tests are phase separation and total energy decrease with different temperature values. The linear stability analysis shows a good agreement between the exact and numerical solutions with an optimal value s. Various computational experiments confirm that the proposed splitting parameter estimation gives stable numerical results. Copyright © 2016 John Wiley &amp; Sons, Ltd.

Fourier Spectral Methods for Some Linear Stochastic Space-Fractional Partial Differential Equations

Fourier spectral methods for solving some linear stochastic space-fractional partial differential equations perturbed by space-time white noises in the one-dimensional case are introduced and analysed. The space-fractional derivative is defined by using the eigenvalues and eigenfunctions of the Laplacian subject to some boundary conditions. We approximate the space-time white noise by using piecewise constant functions and obtain the approximated stochastic space-fractional partial differential equations. The approximated stochastic space-fractional partial differential equations are then solved by using Fourier spectral methods. Error estimates in the L 2 -norm are obtained, and numerical examples are given.

High-order and mass conservative methods for the conservative Allen–Cahn equation

The conservative Allen–Cahn (AC) equation has been studied analytically and numerically. Our mathematical analysis and numerical experiment, however, show that previous numerical methods are not second-order accurate in time and/or do not conserve the initial mass. The aim of this paper is to propose high-order and mass conservative methods for solving the conservative AC equation. In the methods, we discretize the conservative AC equation by using a Fourier spectral method in space and first-, second-, and third-order implicit–explicit Runge–Kutta schemes in time. We show that the methods inherit the mass conservation. Numerical experiments are presented demonstrating the accuracy and efficiency of proposed methods.

Tracing Multiple Solution Branches for Nonlinear Ordinary Differential Equations: Chebyshev and Fourier Spectral Methods and a Degree-Increasing Spectral Homotopy [DISH

Nonlinear differential and integral equations often have multiple solution branches. Reliable sofware to find all roots of a system of polynomial equations, as generated by standard discretizations for differential equations with polynomial nonlinearity, are now widely available. It is now feasible to pursue a global, all-branches attack on differential and integral equations. Unfortunately, both the number of solutions and computational cost grow exponentially fast with N, the number of degrees of freedom. And what if the nonlinearity is not polynomial? And how to continue from small-N to large N? Here, we show that Chebyshev and Fourier Petrov---Galerkin methods are "N-minimizing", but especially with exploitation of cryptoperiodicity, parity and other symmetries, basis recombination, and unconventional Gegenbauer polynomial Petrov---Galerkin weights. We also show that Chebyshev---Pade technology is efficient at "polynomializing" transcendental nonlinearity. If the small-N solution is sufficiently accurate, continuation to larger N is a simple matter of initializing Newton's iteration for large N with the solution for small N ("modal persistence"). When this fails, we introduce Newton-DISH: a Degree-Increasing Spectral Homotopy which is slower but more resilient. The major limitation is that present-day all-branches polynomial system solvers are limited, due to both operation count and memory storage, to rather modest N, typically 4---20, which we dub $$N_{feasible}$$Nfeasible. If the interesting branches can be resolved at least crudely by $$N_{feasible}$$Nfeasible degrees of freedom, then the solutions can be refined to high accuracy by DISH. However, solution branches that demand $$N > N_{feasible}$$N>Nfeasible for even a poor approximation lie outside the scope of the strategy described here. Our theme is not that the "all-roots" strategy is universally applicable, but rather that its domain of usefulness is greatly expanded by using spectral methods in lieu of low order finite differences and finite elements. We demonstrate the power of the polynomial-solver and Chebyshev partnership by finding a hitherto unsuspected second branch (smooth and real-valued) of the Sag-Haselgrove nonlinear integral equation, which has been studied for more than half a century.

Simulating infinite vortex lattices in superfluids

We present an efficient framework to numerically treat infinite periodic vortex lattices in rotating superfluids described by the Gross–Pitaevskii theory. The commonly used split-step Fourier (SSF) spectral methods are inapplicable to such systems as the standard Fourier transform does not respect the boundary conditions mandated by the magnetic translation group. We present a generalisation of the SSF method which incorporates the correct boundary conditions by employing the so-called magnetic Fourier transform. We test the method and show that it reduces to known results in the lowest-Landau-level regime. While we focus on rotating scalar superfluids for simplicity, the framework can be naturally extended to treat multicomponent systems and systems under more general ‘synthetic’ gauge fields.

Formation and evolution of flapping and ballooning waves in magnetospheric plasma sheet

By adopting Lembege & Pellat’s 2D plasma-sheet model, we investigate the flankward flapping motion and Sunward ballooning propagation driven by an external source (e.g., magnetic reconnection) produced initially at the sheet center. Within the ideal MHD framework, we adopt the WKB approximation to obtain the Taylor–Goldstein equation of magnetic perturbations. Fourier spectral method and Runge–Kutta method are employed in numerical simulations, respectively, under the flapping and ballooning conditions. Studies expose that the magnetic shears in the sheet are responsible for the flapping waves, while the magnetic curvature and the plasma gradient are responsible for the ballooning waves. In addition, the flapping motion has three phases in its temporal development: fast damping phase, slow recovery phase, and quasi-stabilized phase; it is also characterized by two patterns in space: propagating wave pattern and standing wave pattern. Moreover, the ballooning modes are gradually damped toward the Earth, with a wavelength in a scale size of magnetic curvature or plasma inhomogeneity, only 1–7% of the flapping one; the envelops of the ballooning waves are similar to that of the observed bursty bulk flows moving toward the Earth.

A Legendre–Fourier spectral method with exact conservation laws for the Vlasov–Poisson system

We present the design and implementation of an L2-stable spectral method for the discretization of the Vlasov- Poisson model of a collisionless plasma in one space and velocity dimension. The velocity and space dependence of the Vlasov equation are resolved through a truncated spectral expansion based on Legendre and Fourier basis functions, respectively. The Poisson equation, which is coupled to the Vlasov equation, is also resolved through a Fourier expansion. The resulting system of ordinary differential equation is discretized by the implicit second-order accurate Crank-Nicolson time discretization. The non-linear dependence between the Vlasov and Poisson equations is iteratively solved at any time cycle by a Jacobian-Free Newton-Krylov method. In this work we analyze the structure of the main conservation laws of the resulting Legendre-Fourier model, e.g., mass, momentum, and energy, and prove that they are exactly satisfied in the semi-discrete and discrete setting. The L2-stability of the method is ensured by discretizing the boundary conditions of the distribution function at the boundaries of the velocity domain by a suitable penalty term. The impact of the penalty term on the conservation properties is investigated theoretically and numerically. An implementation of the penalty term that does not affect the conservation of mass, momentum and energy, is also proposed and studied. A collisional term is introduced in the discrete model to control the filamentation effect, but does not affect the conservation properties of the system. Numerical results on a set of standard test problems illustrate the performance of the method.