Fourth-order Finite Difference Scheme Research Articles

A phase velocity preserving fourth-order finite difference scheme for the Helmholtz equation with variable wavenumber

Numerically solving the Helmholtz equation with large wavenumbers can be challenging due to the highly oscillatory nature of the solution. This paper proposes a 3-point finite difference scheme for numerically solving the 1D Helmholtz equation with variable wavenumber. The proposed scheme preserves the phase velocity, making it well-suited for problems with large wavenumbers. Convergence analysis shows that the proposed scheme is guaranteed to have fourth-order accuracy. Numerical results demonstrate that the proposed scheme maintains high accuracy even for problems with large wavenumbers.

A Five-Step Block Method Coupled with Symmetric Compact Finite Difference Scheme for Solving Time-Dependent Partial Differential Equations

In this article, we present a five-step block method coupled with an existing fourth-order symmetric compact finite difference scheme for solving time-dependent initial-boundary value partial differential equations (PDEs) numerically. Firstly, a five-step block method has been designed to solve a first-order system of ordinary differential equations that arise in the semi-discretisation of a given initial boundary value PDE. The five-step block method is derived by utilising the theory of interpolation and collocation approaches, resulting in a method with eighth-order accuracy. Further, characteristics of the method have been analysed, and it is found that the block method possesses A-stability properties. The block method is coupled with an existing fourth-order symmetric compact finite difference scheme to solve a given PDE, resulting in an efficient combined numerical scheme. The discretisation of spatial derivatives appearing in the given equation using symmetric compact finite difference scheme results in a tridiagonal system of equations that can be solved by using any computer algebra system to get the approximate values of the spatial derivatives at different grid points. Two well-known test problems, namely the nonlinear Burgers equation and the FitzHugh-Nagumo equation, have been considered to analyse the proposed scheme. Numerical experiments reveal the good performance of the scheme considered in the article.

A linearlized mass-conservative fourth-order block-centered finite difference method for the semilinear Sobolev equation with variable coefficients

In this paper, a two-dimensional variable-coefficients semilinear Sobolev equation under Neumann boundary condition is considered, and a novel Crank–Nicolson type linearized fourth-order block-centered finite difference scheme which preserves mass is developed and analyzed. Under local Lipschitz continuous assumption on the nonlinear source term, second-order temporal and fourth-order spatial accuracy are rigorously proved for the primal scalar variable p, its gradient u and its flux q simultaneously. Then, stability under a rough time stepsize condition is proved following the convergence results. Numerical experiments are presented to confirm the theoretical conclusions and well performance of the proposed scheme.

Two fast and unconditionally stable finite difference methods for Riesz fractional diffusion equations with variable coefficients

In this paper, for variable coefficient Riesz fractional diffusion equations in one and two dimensions, we first design a second-order implicit difference scheme by using the Crank-Nicolson method and a fractional centered difference formula for time and space variables, respectively. With the compact operator acting on, a novel fourth-order finite difference scheme is subsequently constructed. Solvability, stability and convergence of these schemes are theoretically analyzed. For these discretized linear systems, the fast implementation with preconditioners based on sine transform is proposed, which has the computational complexity of O(nlog⁡n) per iteration and the memory requirement of O(n), where n represents the total number of the spatial grid nodes. Finally, numerical experiments are performed to illustrate the preciseness and effectiveness of these new techniques.

Compact 9-point finite difference methods with high accuracy order and/or M-matrix property for elliptic cross-interface problems

In this paper we develop finite difference schemes for elliptic problems with piecewise continuous coefficients that have (possibly huge) jumps across fixed internal interfaces. In contrast with such problems involving one smooth non-intersecting interface, that have been extensively studied, there are very few papers addressing elliptic interface problems with intersecting interfaces of coefficient jumps. It is well known that if the values of the permeability in the four subregions around a point of intersection of two such internal interfaces are all different, the solution has a point singularity that significantly affects the accuracy of the approximation in the vicinity of the intersection point. In the present paper we propose a fourth-order 9-point finite difference scheme on uniform Cartesian meshes for an elliptic problem whose coefficient is piecewise constant in four rectangular subdomains of the overall two-dimensional rectangular domain. Moreover, for the special case when the intersecting point of the two lines of coefficient jumps is a grid point, such a compact scheme, involving relatively simple formulas for computation of the stencil coefficients, can even reach sixth order of accuracy. Furthermore, we show that the resulting linear system for the special case has an M-matrix, and prove the theoretical sixth order convergence rate using the discrete maximum principle. Our numerical experiments demonstrate the sixth (for the special case) and at least fourth (for the general case) accuracy orders of the proposed schemes. In the general case, we derive a compact third-order finite difference scheme, also yielding a linear system with an M-matrix. In addition, using the discrete maximum principle, we prove the third order convergence rate of the scheme for the general elliptic cross-interface problem.

A new approach for numerical solution of Kuramoto-Tsuzuki equation

A fourth-order finite difference scheme for the Kuramoto-Tsuzuki equation is considered. The theoretical properties of the proposed scheme, such as the existence, uniqueness and the consistency errors are analyzed. The error estimates in a discrete maximum-norm show that the convergence rates of the difference scheme are of order O(h4+k2). Some numerical experiments are reported to confirm the advantages of the proposed difference scheme by comparing it with other existing recent numerical methods.

Exploration of ternary-hybrid nanofluid experiencing Coriolis and Lorentz forces: case of three-dimensional flow of water conveying carbon nanotubes, graphene, and alumina nanoparticles

ABSTRACT There is a rising industrial need to enhance the thermal properties of fluid as they flow over surfaces. Practical applications include coolants in electronics, heat exchangers and heat transfer liquids. This study communicates flow and heat transfer in magnetohydrodynamic ternary hybrid nanofluid over a rotating three-dimensional surface with the impact of suction velocity. The transport equations are transformed from partial into ordinary differential equations and subsequently solved numerically using the bvp4c solver, which executes the fourth-order three-stage Lobatto IIIa formula finite difference scheme. The impacts of the developmental factors on the velocity and temperature distribution are depicted and discussed using various graphics. The observed results are in great agreement with relevant published studies in the literature under various limiting conditions. It is worth concluding that the stretching parameter decelerates the flow of the ternary hybrid nanofluid in the x direction and lowers the surface temperature, whereas there is enhancement of the velocity field in the y direction with growth in the stretching parameter.

Stagnation point flow of chemically reactive nanofluid due to the curved stretching surface with modified Fourier and Fick theories

With effective thermal performances and stability, the nanomaterials are intended to be the focus of investigators due to applications in energy systems, thermal extrusions, pharmaceutical processes, diagnoses and therapies, domestic refrigerators, bio-medical technologies, nuclear processes, antibacterial activities, etc. This research addresses thermal transport in stagnation point flow of nanofluid over a curved surface with nonlinear radiative effects and chemical reaction effects. Basic mass and momentum conservation laws are used to model the flow problem, whereas Cattaneo–Christov theories are used in thermal diffusion relations to model the heat and concentration equations. The similarity variables transformed the governing partial differential equations and the resulting equations are numerically simulated by the fourth-order finite difference scheme. The effective change in velocity, heat transfer rate, and mass fluctuation are reported due to effective values of parameters. It is noted that velocity, temperature, and concentration are affected by the curvature of the surface.

New approach on conventional solutions to nonlinear partial differential equations describing physical phenomena

In current study, the modified variational iteration algorithm-I is investigated in the form of the analytical and numerical treatment of different types of nonlinear partial differential equations modelling physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to two processes: diffusion and convection. In this modified algorithm, a supplementary parameter which guarantees a faster convergence rate is presented. Appropriate values of the additional parameter are obtained after minimization of properly defined error functions. The obtained results are compared against the exact solutions as well as against numerical solutions generated by the compact finite-difference method, fourth-order finite difference scheme, the second kind Chebyshev wavelets, DQ Chebyshev, and DQ Lagrange methods to show the efficiency, precision and implementation of the method. The proposed algorithm is considered as a very good technique for solving practical problems resulting in different fields of applied physical sciences and engineering.

Analysis and application of a spatial fourth-order finite difference scheme for the Ziolkowski's PML model

Perfectly Matched Layer (PML) technique is an effective tool to reduce the unbounded wave propagation problem to a bounded domain problem. In this paper, we develop and analyze a fourth-order in space finite difference scheme for solving the Ziolkowski's PML model. Though fourth-order in space schemes have been widely used in solving Maxwell's equations, few papers provide a rigorous convergence analysis. One of the major contributions here is to fill this gap by proving the fourth-order pointwise convergence of the scheme. Numerical results are presented to justify our analysis, and demonstrate the effectiveness of this PML model in absorbing outgoing waves.

Higher order compact finite difference schemes for steady and transient solutions of space–time neutron diffusion model

One of the most difficult tasks in the numerical simulation of nuclear reactors is to find solution of a system arising from spatial discretization of multidimensional multigroup neutron diffusion equations by finite difference method. The big disadvantage is that it requires smaller mesh spacing. Using smaller mesh spacing produces reasonable results but it consumes huge computational effort. In this work, we overcome this flaw by constructing higher order compact finite difference schemes to approximate the neutron leakage terms. The technique is based on using concentrated points around a central point instead of minimizing the mesh spacing taking into account the location of the diffusion coefficients. Beside the accuracy of the proposed schemes, the dimension of the coefficient matrix and therefore the complexity of the system still the same. Five point stencils with equally distance h in each direction are used to produce more than one of standard central difference formulas to be combined producing the fourth order finite difference scheme with order of accuracy O(h4). Also, seven point stencils are used to get the sixth order scheme with accuracy O(h6). The proposed fourth and sixth order schemes are highlighted through two theories. The maximum error norms and the convergence rates of the proposed schemes are discussed. For the time discretization, multi-step differential transform method MDTM is employed to solve the neutron diffusion model. To illustrate the applicability and accuracy of the proposed schemes, results are tested on some multidimensional homogenous and heterogeneous benchmark problems. Our static normalized assembly power densities that are obtained by dividing the whole core into just few dozens or hundreds of fuel assemblies are compared with the Improved SIDK, FLUENT and BRBFCM methods that divide the core into many thousands of fuel assemblies. It is proved that the obtained results are very close to those methods.

High order semi-implicit weighted compact nonlinear scheme for the full compressible Euler system at all Mach numbers

A high order semi-implicit weighted compact nonlinear scheme (WCNS) is presented to solve the full compressible Euler equations at all Mach numbers. To avoid the stringent acoustic CFL restriction for an explicit time discretization method, the pressure splitting methodology is introduced to split the Euler equations into nonstiff and stiff terms. The nonstiff and stiff terms are treated explicitly and implicitly, respectively, based on the high order IMEX Runge-Kutta time discretization method that is asymptotic preserving and asymptotically accurate in the zero Mach number limit. A fifth-order WCNS and fourth-order centered finite difference schemes with zero numerical viscosity are used for the spatial discretization. Numerical tests in one and two space dimensions are displayed to demonstrate the performance of the high order semi-implicit WCNS in compressible and incompressible regimes. Comparisons with the third-order explicit weighted essentially non-oscillatory (WENO) scheme are also made in order to better assess the proposed scheme.

An Optimal Fourth-Order Finite Difference Scheme for the Helmholtz Equation Based on the Technique of Matched Interface Boundary

In this paper, a new optimal fourth-order 21-point finite difference scheme is proposed to solve the 2D Helmholtz equation numerically, with the technique of matched interface boundary (MIB) utilized to treat boundary problems. For the approximation of Laplacian, two sets of fourth-order difference schemes are derived firstly based on the Taylor formula, with a total of 21 grid points involved. Then, a weighted combination of the two schemes is employed in order to reduce the numerical dispersion, and the weights are determined by minimizing the dispersion. Similarly, for the discretization of the zeroth-order derivative term, a weighted average of all the 21 points is implemented to obtain the fourth-order accuracy. The new scheme is noncompact; hence, it encounters great difficulties in dealing with the boundary conditions, which is crucial to the order of convergence. To tackle this issue, the matched interface boundary (MIB) method is employed and developed, which is originally used to accommodate free edges in the discrete singular convolution analysis. Convergence analysis and dispersion analysis are performed. Numerical examples are given for various boundary conditions, which show that new scheme delivers a fourth order of accuracy and is efficient in reducing the numerical dispersion as well.

Numerical simulation of differential-difference equations with small delay in convective term and reaction term

In this paper, we suggested a numerical scheme for solving singularly perturbed differential-difference equation with small shift. First, Taylor series used to replace the given problem as singularly perturbed boundary value problem and then subsequently a fourth order finite difference scheme is employed to solve this problem. Convergence of the method is evaluated. By considering numerical experiments, the effect of small shift on the boundary layer solution of the problem is demonstrated.

High order accurate in time, fourth order finite difference schemes for the harmonic mapping flow

In this paper, a fully discrete numerical scheme is proposed and analyzed for the harmonic mapping flow, with the fourth order spatial accuracy and higher than third order temporal accuracy. The fourth order spatial accuracy is realized via a long stencil finite difference, and the boundary extrapolation is implemented by making use of higher order Taylor expansion. Meanwhile, the high order (third or fourth order) temporal accuracy is based on a semi-implicit algorithm, which uses a combination of explicit Adams–Bashforth extrapolation for the nonlinear terms and implicit Adams–Moulton interpolation for the viscous diffusion term, with the corresponding integration formula coefficients. Both the consistency, linearized stability analysis and optimal rate convergence estimate (in the ℓ∞(0,T;ℓ2)∩ℓ2(0,T;Hh1) norm) are provided. A few numerical examples are also presented in this article.

A Numerical Discussion for the European Put Option Model

The Black-Scholes equations have been increasingly popular over the last three decades since they provide more practical information for optional behaviours. Therefore, effective methods have been needed to analyze these models. This study will focus mainly on investigating the behavior of the Black-Scholes equation for the European put option pricing model. To achieve this, numerical solutions of the Black-Scholes European option pricing model are produced by three combined methods. Spatial discretization of the Black-Scholes model is performed using a fourth-order finite difference (FD4) scheme that allows a highly accurate approximation of the solutions. For the time discretization, three numerical techniques are proposed: a strong-stability preserving Runge Kutta (SSPRK3), a fourth-order Runge Kutta (RK4) and a one-step method. The results produced by the combined methods have been compared with available literature and the exact solution.

A high-order linearized difference scheme preserving dissipation property for the 2D Benjamin-Bona-Mahony-Burgers equation

In this paper, a fourth-order finite difference scheme that preserves the dissipation of energy for solving the 2D Benjamin-Bona-Mahony-Burgers (BBMB) equation is proposed. The scheme is three-level in time and linear-implicit. The solvability, unconditional stability and convergence are rigorously proved by the discrete energy method. The rate of convergence is of O(τ2+h4). Numerical examples are given to confirm the good accuracy and the effectiveness of the present scheme for handling the 2D homogeneous and inhomogeneous BBMB equations.

A generalized optimal fourth-order finite difference scheme for a 2D Helmholtz equation with the perfectly matched layer boundary condition

A crucial part of successful wave propagation related inverse problems is an efficient and accurate numerical scheme for solving the seismic wave equations. In particular, the numerical solution to a multi-dimensional Helmholtz equation can be troublesome when the perfectly matched layer (PML) boundary condition is implemented. In this paper, we present a general approach for constructing fourth-order finite difference schemes for the Helmholtz equation with PML in the two-dimensional domain based on point-weighting strategy. Particularly, we develop two optimal fourth-order finite difference schemes, optimal point-weighting 25p and optimal point-weighting 17p. It is shown that the two schemes are consistent with the Helmholtz equation with PML. Moreover, an error analysis for the numerical approximation of the exact wavenumber is provided. Based on minimizing the numerical dispersion, we implement the refined choice strategy for selecting optimal parameters and present refined point-weighting 25p and refined point-weighting 17p finite difference schemes. Furthermore, three numerical examples are provided to illustrate the accuracy and effectiveness of the new methods in reducing numerical dispersion.

Solving SPDDE using fourth order numerical method

In this paper we present a fourth order numerical method to solve singularly perturbed differential-difference equations. The solution of this problem exhibits layer behaviour at one end. A fourth order finite difference scheme on a uniform mesh is developed. The effect of delay and advance parameters on the boundary layer(s) has also been analyzed and depicted in graphs. The applicability of the proposed scheme is validated by implementing it on model examples. To show the accuracy of the method, the results are presented in terms of maximum absolute errors.

Fast algorithms for the electromagnetic scattering by partly covered cavities

We study the electromagnetic scattering by the partly covered cavity with high wave numbers, which is governed by a Helmholtz equation with the transparent boundary condition. First, we develop a fast high-order algorithm for the scattering by the partly covered cavity filled with the homogeneous medium. The Helmholtz equation is discretized using a fourth-order finite-difference scheme and then reduced into a small aperture system by the fast Fourier transformation and Gaussian elimination. The existence and uniqueness of the numerical solution are given. By means of the immersed interface method, we also propose a fast high-order method for solving the scattering by the partly covered cavity with inhomogeneous media. A preconditioned iterative method based on Krylov subspace is applied to accelerate the convergence. Numerical experiments with different types of media are presented to demonstrate the efficiency and validity of the proposed methods.