Spectral Finite Difference Method Research Articles

Unsteady oblique stagnation point flow with improved pressure field and fractional Cattaneo–Christov model by finite difference-spectral method

Unsteady oblique stagnation point flow with improved pressure field and fractional Cattaneo–Christov model by finite difference-spectral method

A Hybrid Spectral-Finite Difference Method for Numerical Pricing of Time-Fractional Black–Scholes Equation

A Hybrid Spectral-Finite Difference Method for Numerical Pricing of Time-Fractional Black–Scholes Equation

Magnetofluid unsteady electroosmotic flow of Jeffrey fluid at high zeta potential in parallel microchannels

Abstract In this article, the magnetofluid unsteady electroosmotic flow (EOF) of Jeffrey fluid with high zeta potential is studied by using the Chebyshev spectral method and the finite difference method. By comparing the potential distribution and velocity distribution obtained by the Chebyshev spectral method and finite difference method, it is concluded that the Chebyshev spectral method has higher precision and less computation. Then the numerical solution obtained by the Chebyshev spectral method is used to analyze the flow characteristics of Jeffrey fluid at high zeta potential. The results show that the velocity of Jeffrey fluid increases with the increase of the wall zeta potential and electric field intensity. The oscillation amplitude of velocity distribution increases with the increase of relaxation time, but decreases with the increase of retardation time. With the increase of Hartmann number, the velocity first increases and then decreases. The positive pressure gradient promotes the flow of fluid, and the reverse pressure gradient impedes the flow of fluid.

Physics‐Informed Neural Networks (PINNs) for Wave Propagation and Full Waveform Inversions

AbstractWe propose a new approach to the solution of the wave propagation and full waveform inversions (FWIs) based on a recent advance in deep learning called physics‐informed neural networks (PINNs). In this study, we present an algorithm for PINNs applied to the acoustic wave equation and test the method with both forward models and FWI case studies. These synthetic case studies are designed to explore the ability of PINNs to handle varying degrees of structural complexity using both teleseismic plane waves and seismic point sources. PINNs' meshless formalism allows for a flexible implementation of the wave equation and different types of boundary conditions. For instance, our models demonstrate that PINN automatically satisfies absorbing boundary conditions, a serious computational challenge for common wave propagation solvers. Furthermore, a priori knowledge of the subsurface structure can be seamlessly encoded in PINNs' formulation. We find that the current state‐of‐the‐art PINNs provide good results for the forward model, even though spectral element or finite difference methods are more efficient and accurate. More importantly, our results demonstrate that PINNs yield excellent results for inversions on all cases considered and with limited computational complexity. We discuss the current limitations of the method with complex velocity models as well as strategies to overcome these challenges. Using PINNs as a geophysical inversion solver offers exciting perspectives, not only for the full waveform seismic inversions, but also when dealing with other geophysical datasets (e.g., MT, gravity) as well as joint inversions because of its robust framework and simple implementation.

Taming Hyperchaos with Exact Spectral Derivative Discretization Finite Difference Discretization of a Conformable Fractional Derivative Financial System with Market Confidence and Ethics Risk

Four discrete models, using the exact spectral derivative discretization finite difference (ESDDFD) method, are proposed for a chaotic five-dimensional, conformable fractional derivative financial system incorporating ethics and market confidence. Since the system considered was recently studied using the conformable Euler finite difference (CEFD) method and found to be hyperchaotic, and the CEFD method was recently shown to be valid only at fractional index α=1, the source of the hyperchaos is in question. Through numerical experiments, illustration is presented that the hyperchaos previously detected is, in part, an artifact of the CEFD method, as it is absent from the ESDDFD models.

Galerkin spectral method for a multi-term time-fractional diffusion equation and an application to inverse source problem

<abstract><p>In this paper, we employ the Galerkin spectral method to handle a multi-term time-fractional diffusion equation, and investigate the numerical stability and convergence of the proposed method. In addition, we find an interesting application of the Galerkin spectral method to solving an inverse source problem efficiently from the noisy final data in a general bounded domain, and the uniqueness and the ill-posedness for the inverse problem are proved based on expression of the solution. Furthermore, we compare the calculation results of spectral method and finite difference method without any regularization method, and get a norm estimate of the coefficient matrix of a spectral method discretized. And for that we conclude that the spectral method itself can act as a regularization method for some inverse problem (such as inverse source problem). Finally, several numerical examples are used to illustrate the effectiveness and accuracy of the method.</p></abstract>

A Legendre spectral-finite difference method for Caputo–Fabrizio time-fractional distributed-order diffusion equation

A Legendre spectral-finite difference method for Caputo–Fabrizio time-fractional distributed-order diffusion equation

A Finite Difference-Spectral Method for Solving the European Call Option Black–Scholes Equation

In this paper, we present a novel technique based on backward-difference method and Galerkin spectral method for solving Black–Scholes equation. The main propose of this method is to reduce the solution of this problem to the solution of a system of algebraic equations. The convergence order of the proposed method is investigated. Also, we provide numerical experiment to show the validity of proposed method.

A parallel in time/spectral collocation combined with finite difference method for the time fractional differential equations

In this article, we consider the numerical solution for the time fractional differential equations (TFDEs). We propose a parallel in time method, combined with a spectral collocation scheme and the finite difference scheme for the TFDEs. The parallel in time method follows the same sprit as the domain decomposition that consists in breaking the domain of computation into subdomains and solving iteratively the sub-problems over each subdomain in a parallel way. Concretely, the iterative scheme falls in the category of the predictor-corrector scheme, where the predictor is solved by finite difference method in a sequential way, while the corrector is solved by computing the difference between spectral collocation and finite difference method in a parallel way. The solution of the iterative method converges to the solution of the spectral method with high accuracy. Some numerical tests are performed to confirm the efficiency of the method in three areas: (i) convergence behaviors with respect to the discretization parameters are tested; (ii) the overall CPU time in parallel machine is compared with that for solving the original problem by spectral method in a single processor; (iii) for the fixed precision, while the parallel elements grow larger, the iteration number of the parallel method always keep constant, which plays the key role in the efficiency of the time parallel method.

Computational heat transfer with spectral graph theory: Quantitative verification

The objective of this paper is to quantify the precision of a novel approach for computational heat transfer modeling based on spectral graph theory. Two benchmark heat transfer problems with planar boundaries, for which exact analytical solutions are available, are used to determine the precision of temperature predictions obtained from spectral graph, finite difference (one-dimensional), and finite element (three-dimensional) methods. These studies show that the spectral graph approach captures the temperature trends in the benchmark case studies with error in the range of 2% to 10% depending on the location in the body. These verification studies also provide an approach to calibrate the numerical parameters in the new method. The spectral graph approach is applied for predicting the thermal history of a complex three-dimensional additive manufactured (3D printed) part. The temperature trends in a 50-layer part are computed 2.3 times faster than a commercial finite-element software package, and the results differ by less than 7.5%. Further improvements in the computational speed of the spectral graph approach are expected through code optimization and parallelization. This work has far-reaching practical implications for predicting thermal-induced defects in a variety of manufacturing processes including casting and additive manufacturing.

Solving a fractional parabolic-hyperbolic free boundary problem which models the growth of tumor with drug application using finite difference-spectral method

In this paper, a free boundary problem modelling the growth of tumor is considered. The model includes two reaction-diffusion equations modelling the diffusion of nutrient and drug in the tumor and three hyperbolic equations describing the evolution of three types of cells (i.e. proliferative cells, quiescent cells and dead cells) considered in the tumor. Due to the fact that in the real situation, the subdiffusion of nutrient and drug in the tumor can be found, we have changed the reaction-diffusion equations to the fractional ones to consider other conditions and study a more general and reliable model of tumor growth. Since it is important to solve a problem to have a clear vision of the dynamic of tumor growth under the effect of the nutrient and drug, we have solved the fractional free boundary problem. We have solved the fractional parabolic equations employing a combination of spectral and finite difference methods and the hyperbolic equations are solved using characteristic equation and finite difference method. It is proved that the presented method is unconditionally convergent and stable to be sure that we have a correct vision of tumor growth dynamic. Finally, by presenting some numerical examples and showing the results, the theoretical statements are justified.

Fast 3D Poisson solvers in elliptical conducting pipe for space-charge simulation

Author(s): Qiang, Ji | Abstract: Space-charge effects play an important role in high intensity accelerators. These effects can be studied self-consistently by solving the Poisson equation with the dynamically evolved charge density distribution subject to appropriate boundary conditions. In this paper, two computationally efficient methods are proposed to solve the Poisson equation inside an elliptical perfectly conducting pipe. One method uses a spectral method and the other uses a spectral finite difference method. The former method has a high accuracy and the latter one has a computational complexity of O(Nlog(N)), where N is the total number of unknowns. These methods implemented in a beam dynamics tracking code enable the fast simulation of space-charge effects in an accelerator with an elliptical conducting pipe.

A Legendre spectral finite difference method for the solution of non-linear space-time fractional Burger’s–Huxley and reaction-diffusion equation with Atangana–Baleanu derivative

Abstract In this research, we have solved non-linear reaction-diffusion equation and non-linear Burger’s–Huxley equation with Atangana Baleanu Caputo derivative. We developed a numerical approximation for the ABC derivative of Legendre polynomial. A difference scheme is applied to deal with fractional differential term in the time direction of differential equation. We applied Legendre spectral method to deal with unknown function and spatial ABC derivatives. A formulation to deal with Dirichlet boundary condition is also included. After applying this spectral method our problem reduces to a system of fractional partial differential equation. To solve this system we developed finite difference scheme by which our FPDEs system reduces to a system of algebraic equations. Taking the help of initial conditions we solve this algebraic system and find the value of unknowns, To demonstrate the effectiveness and validity of our proposed method some numerical examples are also presented. We compare our obtained numerical results with the analytical results.

On the numerical evaluation for studying the fractional KdV, KdV-Burgers and Burgers equations

This paper is devoted to present an accurate numerical procedure to solve fractional (Caputo sense) Korteweg-de Vries, Korteweg-de Vries-Burgers and Burgers equations by using the spectral Chebyshev collocation method and finite difference method (FDM). The proposed problem is reduced to a system of ODEs with the help of the properties of Chebyshev polynomials of the third kind. This system is solved by using the FDM. Some theorems about the convergence analysis are stated and proved. A numerical simulation and a comparison with the previous work are presented.

The stabilization of high-order multistep schemes for the Laguerre one-way wave equation solver

This paper considers spectral-finite difference methods of a high-order of accuracy for solving the one-way wave equation using the Laguerre integral transform with respect to time as the base. In order to provide a high spatial accuracy and stability, the Richardson method can be employed. However such an approach requires high computer costs, therefore we consider alternative algorithms based on the Adams multistep schemes. To reach the stability for the one-way equation, the stabilizing procedures using the spline interpolation were developed. This made it possible to efficiently implement a predictor–corrector type method thus decreasing computer costs. The stability and accuracy of the procedures proposed have been studied, based on the implementation of the migration algorithm within a problem of seismic prospecting.

An effective comparison involving a novel spectral approach and finite difference method for the Schrödinger equation involving the Riesz fractional derivative in the quantum field theory

This paper displays the approach of the time-splitting Fourier spectral (TSFS) technique for the linear Riesz fractional Schrodinger equation (RFSE) in the semi-classical regime. The splitting technique is shown to be unconditionally stable. Further a suitable implicit finite difference discretization of second order has been manifested for the RFSE where the Riesz derivative has been discretized via an approach of fractional centered difference. Moreover the stability analysis for the implicit scheme has also been presented here via von Neumann analysis. The L2-norm and $L^{\infty}$ -norm errors are calculated for $\vert u(x,t)\vert^{2}$ , Re $(u(x,t))$ and Im $(u(x,t))$ for various cases. The results obtained by the methods are further tabulated for the absolute errors for $\vert u(x,t)\vert^{2}$ . Furthermore the graphs are depicted showing comparison of $\vert u(x,t)\vert^{2}$ by both techniques. The derivatives are taken here in the context of the Riesz fractional sense. Apart from that, the comparative study put forth in the following section via tables and graphs between the implicit second-order finite difference method (IFDM) and the TSFS method is for the purpose of investigating the efficiency of the results obtained. Moreover the stability analysis of the presented techniques manifesting their unconditional stability makes the proposed approach more competing and accurate.

A REDUCED-ORDER EXTRAPOLATION SPECTRAL-FINITE DIFFERENCE SCHEME BASED ON THE POD METHOD FOR 2D SECOND-ORDER HYPERBOLIC EQUATIONS

In this study, a reduced-order extrapolation spectral-finite difference (ROESFD) scheme based on the proper orthogonal decomposition (POD) method is set up for the two-dimensional (2D) second-order hyperbolic equations. First, the classical spectral-finite difference (CSFD) method for the 2D second-order hyperbolic equations and its stability, convergence, and flaw are introduced. Then, a new ROESFD scheme that has very few degrees of freedom but holds sufficiently high accuracy is set up by the POD method and its implementation is offered. Finally, three numerical examples are offered to explain the validity of the theoretical conclusion. This implies that the ROESFD scheme is viable and efficient for searching the numerical solutions of the 2D second-order hyperbolic equations.

A fast parallel 3D Poisson solver with longitudinal periodic and transverse open boundary conditions for space-charge simulations

A three-dimensional (3D) Poisson solver with longitudinal periodic and transverse open boundary conditions can have important applications in beam physics of particle accelerators. In this paper, we present a fast efficient method to solve the Poisson equation using a spectral finite-difference method. This method uses a computational domain that contains the charged particle beam only and has a computational complexity of O(Nu(logNmode)), where Nu is the total number of unknowns and Nmode is the maximum number of longitudinal or azimuthal modes. This saves both the computational time and the memory usage of using an artificial boundary condition in a large extended computational domain. The new 3D Poisson solver is parallelized using a message passing interface (MPI) on multi-processor computers and shows a reasonable parallel performance up to hundreds of processor cores.

Numerical Analysis of Secondary Flow in the Narrow Gap of Magnetic Fluid Shaft Seal Using a Spectral Finite Difference Method

ABSTRACTIn the article, attention is focused on a secondary flow in the narrow gap of a magnetic fluid shaft seal, and the secondary flow and circumferential speed of the magnetic fluid are numerically evaluated with variation in magnetic fluid plug shape such as angle of the polar head and seal clearance to be varied, using a spectral finite difference method with a new mapped function. Moreover, the maximum pressure drop is calculated in several patterns of equilibrium magnetic fluid plug, and the influence of the centrifugal force is also discussed.

Verification of a non-hydrostatic dynamical core using the horizontal spectral element method and vertical finite difference method: 2-D aspects

Abstract. The non-hydrostatic (NH) compressible Euler equations for dry atmosphere were solved in a simplified two-dimensional (2-D) slice framework employing a spectral element method (SEM) for the horizontal discretization and a finite difference method (FDM) for the vertical discretization. By using horizontal SEM, which decomposes the physical domain into smaller pieces with a small communication stencil, a high level of scalability can be achieved. By using vertical FDM, an easy method for coupling the dynamics and existing physics packages can be provided. The SEM uses high-order nodal basis functions associated with Lagrange polynomials based on Gauss–Lobatto–Legendre (GLL) quadrature points. The FDM employs a third-order upwind-biased scheme for the vertical flux terms and a centered finite difference scheme for the vertical derivative and integral terms. For temporal integration, a time-split, third-order Runge–Kutta (RK3) integration technique was applied. The Euler equations that were used here are in flux form based on the hydrostatic pressure vertical coordinate. The equations are the same as those used in the Weather Research and Forecasting (WRF) model, but a hybrid sigma–pressure vertical coordinate was implemented in this model. We validated the model by conducting the widely used standard tests: linear hydrostatic mountain wave, tracer advection, and gravity wave over the Schär-type mountain, as well as density current, inertia–gravity wave, and rising thermal bubble. The results from these tests demonstrated that the model using the horizontal SEM and the vertical FDM is accurate and robust provided sufficient diffusion is applied. The results with various horizontal resolutions also showed convergence of second-order accuracy due to the accuracy of the time integration scheme and that of the vertical direction, although high-order basis functions were used in the horizontal. By using the 2-D slice model, we effectively showed that the combined spatial discretization method of the spectral element and finite difference methods in the horizontal and vertical directions, respectively, offers a viable method for development of an NH dynamical core.