Spectral Finite Difference Research Articles

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

An efficient spectral element method for two-dimensional magnetotelluric modeling

We introduce a new efficient spectral element approach to solve the two-dimensional magnetotelluric forward problem based on Gauss–Lobatto–Legendre polynomials. It combines the high accuracy of the spectral technique and the perfect flexibility of the finite element approach, which can significantly improve the calculation accuracy. This method mainly includes two steps: 1) transforming the boundary value problem in the partial differential form into the variational problem in the integral form and 2) solving large symmetric sparse systems based on the combination of incomplete LU factorization and the double conjugate gradient stability algorithm through the spectral element with quadrilateral meshes. We imply the spectral element method on a resistivity half-space model to obtain a simple analytical solution and find that the magnetic field solutions simulated by the spectral element approach matched closely to the exact solutions. The experiment result shows that the spectral element solution has high accuracy with coarse meshes. We further compare the numerical results of the spectral element, finite difference, and finite element approaches on the COMMEMI 2D-1 and smooth models, respectively. The numerical results of the spectral element procedure are highly consistent with the other two techniques. All these comparison results suggest that the spectral element technique can not only give high accuracy for modeling results but also provide more detailed information. In particular, a few nodes are required in this method relative to the finite difference and finite element methods, which can decrease the relative errors. We then deduce that the spectral element method might be an alternative approach to simulate the magnetotelluric responses in two- or three-dimensional structures.

Persistence of dynamic consistency of nonstandard numerical schemes for the Fisher-KPP equation

Finite difference schemes for the Fisher-KPP equation are considered that are constructed using non-local discretization of non-linear terms and standard/nonstandard denominators. Their dynamic consistency with respect to positivity, boundedness, steady-states, and diffusion-front propagation are established. Diffusion-fronts are obtained that inflate, deflate, or remain constant at a geometric rate determined by the time-space step size functional relationship. The persistence of said dynamic consistency in terms of their boundedness, positivity, and determining-diffusion curves at small and large scales is rigorously analyzed and numerically supported. Comparisons are presented for three cases using the Euler, nonstandard finite difference, or exact spectral derivative discretization finite difference denominators. Finally, general measures of robustness in persistence are introduced that incorporate (mesh) scale sizes and are used to rank the robustness of the schemes investigated.

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.

High order approach for solving chaotic and hyperchaotic problems

In this work, the method of Taylor's decomposition on two points is suggested in order to find approximate solutions of chaotic and hyperchaotic initial value problems and to analyze the behaviors of these solutions. Unlike to the classical Taylor's method, the proposed numerical scheme is based on the application of the Taylor's decomposition on two points to the system of nonlinear initial value problems, and as a result an implicit method is obtained. Stability and error analysis of the method are presented, and its high-order accuracy and A-stability are proven. One of the advantages of the proposed method is that it is a stable and very efficient method for chaotic problems as it is an implicit one-step method. The most important advantage of the Taylor's decomposition method is that it has high order accuracy for large step sizes with a simple algorithm compared to other methods. The applicability of the proposed method has been examined in some famous chaotic systems; the Lorenz and Chen systems, and hyperchaotic systems; the Chua and Rabinovich-Fabrikant systems, to emphasize both its accuracy and effectiveness. The accuracy of the proposed method is checked by comparing the calculated results with semi-explicit Adams-Bashforth-Moulton method and ninth order Runge-Kutta method. The calculated results are also compared with multi-stage spectral relaxation method and multi-domain compact finite difference relaxation method. Comparisons have shown that the method is more accurate and efficient than the other mentioned methods for large step sizes. The obtained results are also compared with the theoretical findings and it is shown that the theoretical and numerical results are consistent.

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.

Two combined methods for the global solution of implicit semilinear differential equations with the use of spectral projectors and Taylor expansions

Two combined numerical methods for solving semilinear differential-algebraic equations (DAEs) are obtained and their convergence is proved. The comparative analysis of these methods is carried out and conclusions about the effectiveness of their application in various situations are made. In comparison with other known methods, the obtained methods require weaker restrictions for the nonlinear part of the DAE. Also, the obtained methods enable to compute approximate solutions of the DAEs on any given time interval and, therefore, enable to carry out the numerical analysis of global dynamics of mathematical models described by the DAEs. The examples demonstrating the capabilities of the developed methods are provided. To construct the methods we use the spectral projectors, Taylor expansions and finite differences. Since the used spectral projectors can be easily computed, to apply the methods it is not necessary to carry out additional analytical transformations.

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 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.

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.

A simple implementation of PML for second-order elastic wave equations

When modeling time-domain elastic wave propagation in an unbound space, the standard perfectly matched layer (PML) is straightforward for the first-order partial differential equations (PDEs); by contrast, the PML requires tremendous re-constructions of the governing equations in the second-order PDE form, which is however preferable, because of much less memory and time consumption. Therefore, it is imperative to explore a simple implementation of PML for the second-order system. In this work, we first systematically extend the first-order Nearly PML (NPML) technique into second-order systems, implemented by the spectral element and finite difference time-domain algorithms. It merits the following advantages: the simplicity in implementation, by keeping the second-order PDE-based governing equations exactly the same; the efficiency in computation, by introducing a set of auxiliary ordinary differential equations (ODEs). Mathematically, this PML technique effectively hybridizes the second-order PDEs and first-order ODEs, and locally attenuates outgoing waves, thus efficiently avoid either spatial or temporal global convolutions. Numerical experiments demonstrate that the NPML for the second-order PDE has an excellent absorbing performance for elastic, anelastic and anisotropic media in terms of the absorption accuracy, implementation complexity, and computation efficiency.

A paired spectral-finite difference approach for solving boundary layer flow problems

This study presents an innovative numerical technique for obtaining solutions to systems of partial differential equations. Two previously introduced numerical methods, namely the spectral quasilinearization method (SQLM) (Motsa, in J Appl Math 2013:Article ID 423628, 2013) and the spectral local linearization method (SLLM) (Motsa 2013) have been shown to be efficient methods for solving systems of PDEs but both contain slight limitations. Our proposed method hereinafter referred to as the paired spectral quasilinearization method (PSQLM), seeks to simplify a large system of PDEs by decoupling it into pairs of sub-systems and applying quasilinearization on the pairs in order to linearize the pairs. Crank–Nicolson scheme and spectral collocation method are, respectively, applied to discretize the system of decoupled linearized pairs of PDEs. The PSQLM is described for a general system of PDEs and a numerical experiment is conducted on a system of four differential equations that contain three possible combinations. We compare the different combinations to ascertain the effect of performing different pairings on the accuracy and convergence speed of solutions. We also compare the PSQLM with the SQLM (which solves the system as coupled) and the SLLM (that decouples a system) to test the efficacy of the PSQLM. The observations made from the comparison proves the PSQLM converges faster than the SLLM with very few iterations. It is also shown to use significantly lesser computational time than the SQLM to generate convergent solutions. The accuracy of the PSQLM is also observed to better both the SQLM and SLLM.

Two combined methods for the global solution of implicit semilinear differential equations with the use of spectral projectors and Taylor expansions

Two combined numerical methods for solving semilinear differential-algebraic equations (DAEs) are obtained and their convergence is proved. The comparative analysis of these methods is carried out and conclusions about the effectiveness of their application in various situations are made. In comparison with other known methods, the obtained methods require weaker restrictions for the nonlinear part of the DAE. Also, the obtained methods enable to compute approximate solutions of the DAEs on any given time interval and, therefore, enable to carry out the numerical analysis of global dynamics of mathematical models described by the DAEs. The examples demonstrating the capabilities of the developed methods are provided. To construct the methods we use the spectral projectors, Taylor expansions and finite differences. Since the used spectral projectors can be easily computed, to apply the methods it is not necessary to carry out additional analytical transformations.

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.

Flow past a slippery cylinder: part 2 - elliptic cylinder

Part 2 of this paper is devoted to two-dimensional unsteady flow of a viscous incompressible fluid past an inclined elliptic cylinder subject to impermeability and Navier slip conditions on the surface. As in Part 1, the flow is calculated using two methods: an approximate analytical solution in the form of an asymptotic expansion and a numerical solution based on a spectral-finite difference scheme. The results reveal excellent agreement between the analytical and numerical solutions for small times and moderately large Reynolds numbers. Extensive results focussing on asymmetrical flows are presented for Reynolds numbers 500 and 1000 for various aspect ratios, inclinations, and slip lengths. Comparisons with the no-slip case are also made to elicit the effects of the Navier slip condition. The main findings include a reduction in drag as well as a suppression in vortex shedding.

Flow past a slippery cylinder: part 1—circular cylinder

Part 1 of this two-part paper solves the two-dimensional problem of the unsteady flow of a viscous incompressible fluid past a circular cylinder subject to impermeability and Navier-slip conditions on the surface. The Navier-slip condition is characterized by the slip length, $$\beta $$ . The flow is calculated using two methods. The first takes the form of a double series solution where an expansion is carried out in powers of the time, t, and in powers of the parameter $$\lambda =\sqrt{8t/R}$$ where R is the Reynolds number. This approximate analytical solution is valid for small times following the start of the motion and for large Reynolds numbers. The second method involves a spectral-finite difference procedure for numerically integrating the full Navier–Stokes equations expressed in terms of a stream function and vorticity. Our results demonstrate that for small times and moderately large R the two methods of solution are in excellent agreement. Results are presented for Reynolds numbers 500 and 1000, and comparisons with the no-slip condition are made. The key finding is that a reduction in the drag coefficient results from the Navier-slip condition when compared to the no-slip condition. This reduction increases as $$\beta $$ increases. In addition, the slip condition also leads to suppression in flow separation.