Showing papers on "Finite difference coefficient published in 2022"

A Comparison of Finite Difference and Finite Volume Methods with Numerical Simulations: Burgers Equation Model

Abstract: In this paper, we present an intensive investigation of the finite volume method (FVM) compared to the finite difference methods (FDMs). In order to show the main difference in the way of approaching the solution, we take the Burgers equation and the Buckley–Leverett equation as examples to simulate the previously mentioned methods. On the one hand, we simulate the results of the finite difference methods using the schemes of Lax–Friedrichs and Lax–Wendroff. On the other hand, we apply Godunov’s scheme to simulate the results of the finite volume method. Moreover, we show how starting with a variational formulation of the problem, the finite element technique provides piecewise formulations of functions defined by a collection of grid data points, while the finite difference technique begins with a differential formulation of the problem and continues to discretize the derivatives. Finally, some graphical and numerical comparisons are provided to illustrate and corroborate the differences between these two main methods.

A whole high-accuracy numerical calculation system for the 1D Poisson equation by the interpolation finite difference method

Abstract: The interpolation finite difference method (IFDM) allows free numerical analysis of elliptic partial differential equations over arbitrary domains. Conventionally, in the finite difference method (FDM), the calculation is performed using the second-order accuracy central difference. For engineering problems, second-order accuracy calculations are often sufficient. On the other hand, much research has been carried out to improve the accuracy of numerical calculations. Although there is much research in the FDM field, the development of numerical calculations by the spectral method is decisive in improving the calculation accuracy. Numerical calculations are usually performed by double precision calculations. If double precision calculations ensure 15 significant digits in floating point computing, such numerical calculations will be the ultimate goal to reach. A numerical calculation that does not seem to have an error even though it originally has an error is defined as a virtual error-zero (VE0) calculation. In this paper, we will examine an overall picture of high-accuracy numerical calculation by the IFDM in the numerical calculation of the 1D Poisson equation. It becomes clear that a VE0 calculation is always possible in the numerical calculation method, defined as the compact interpolation finite difference scheme [( m)].

A Hybrid Finite Difference Method for Singularly Perturbed Delay Partial Differential Equations with Discontinuous Coefficient and Source

Abstract: The article presents a hybrid finite difference scheme to solve a singularly perturbed parabolic functional differential equation with discontinuous coefficient and source. The simultaneous presence of deviating argument with a discontinuous source and coefficient makes the problem stiff. The solution of the problem exhibits turning point behaviour across discontinuity as tends to zero. The hybrid scheme presented is a composition of a central difference scheme and a midpoint upwind scheme on a specially generated mesh. At the same time, an implicit finite difference method is used to discretize the time variable. Consistency, stability, and convergence of the presented numerical approach have been investigated. The presented method converges uniformly independent of the perturbation parameter. Numerical results have been presented for two test examples that verify the effectiveness of the scheme.

A novel convenient finite difference method for shallow water waves derived by fifth‐order Kortweg and De‐Vries‐type equation

Abstract: In this research, two new finite difference schemes are derived and presented for estimating a solution to the fifth‐order Kortweg and De‐Vries equation. The global conservation law on any time–space regions which yields a three‐level linear‐implicit algorithm with a diagonal system is exactly preserved by the intended finite difference method. Theoretically verified and numerically proved, the created schemes are unconditionally stable and have the second‐order accuracy both in time and space. The obtained results guarantee that the novel idea offers a new aspect to analyze the wave behavior.

A finite difference based simple iteration method for solving boundary layer flow problems

Abstract: This paper attempts to streamline a novel numerical algorithm termed as spectral simple iteration method with finite difference (SIM-FD). SIM-FD basically works on two principle: (i) nonlinear terms are considered as combination of known and unknown functions at a particular iteration and (ii) In each nonlinear terms of an particular equation, the governing function with higher derivative is set to unkown. Present algorithm is exercised on a model available in literature. It has been shown that the proposed numerical method is capable of producing accurate solutions of nonlinear differential equations by combining relaxation, spectral methods and finite difference. This method seeks to avoid truncation associated with linearization while inclusion of finite difference. This makes the method robust to accommodate coarse boundaries. Solution and residual error analysis are performed to test the convergence and accuracy of the proposed method. Procured results lead to a conclusion that the proposed method is quite efficient in terms of convergence and accuracy.

High-accuracy and high-speed calculation of second-order ODE by the interpolation finite difference method

Abstract: Abstract Numerical calculation of differential equations using the finite difference method (FDM) is entering a new phase. The high-accuracy calculation system of the interpolation FDM (IFDM) enables extremely high-accuracy numerical calculation. One of the higher-order difference schemes of IFDM is defined as the compact interpolation finite difference (CIFD) scheme. According to this finite difference (FD) scheme, it becomes clear that it can be calculated with 15 significant figures in the numerical analysis of the 1D Poisson equation under double precision calculation. Thus, it is possible to perform an apparent error-free calculation. The 1D Poisson equation is a special form of the second-order ordinary differential equation (ODE), but this paper shows that the methods reported thus far are generalized to the general second-order ODE.

Resolving the spurious-state problem in the Dirac equation with finite difference method

Abstract: To solve the Dirac equation with the finite difference method, one has to face up to the spurious-state problem due to the fermion doubling problem when using the conventional central difference formula to calculate the first-order derivative on the equal interval lattices. This problem is resolved by replacing the central difference formula with the asymmetric difference formula, i.e., the backward or forward difference formula. To guarantee the hermitian of the Hamiltonian matrix, the backward and forward difference formula should be used alternatively according to the parity of the wavefunction. This provides a simple and efficient numerical prescription to solve various relativistic problems in the microscopic world.

An Extended Finite Difference Method for Singular Perturbation Problems on a Non-Uniform Mesh

Abstract: Abstract An extended second order finite difference method on a variable mesh is proposed for the solution of a singularly perturbed boundary value problem. A discrete equation is achieved on the non uniform mesh by extending the first and second order derivatives to the higher order finite differences. This equation is solved efficiently using a tridiagonal solver. The proposed method is analysed for convergence, and second order convergence is derived. Model examples are solved by the proposed scheme and compared with available methods in the literature to uphold the method.

A stable class of finite difference scheme for time-fractional partial differential equation

Abstract: Abstract In this article, a numerical study is introduced for solving the fractional partial differential equation (FPDE) arising from electromagnetic waves in dielectric media (EMWDM) by using an efficient class of finite difference methods. The proposed numerical finite difference scheme is based on the Hermite formula. The Caputo's fractional derivatives in time are discretized by a finite difference scheme of order O(k(3-α) ) and O(k(3-β) ), 1<����������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������

Resolving the spurious-state problem in the Dirac equation with the finite-difference method

Abstract: To solve the Dirac equation with the finite-difference method, one has to face the spurious-state problem due to the fermion doubling problem when using the conventional central difference formula to calculate the first-order derivative on equal interval lattices. This problem is resolved by replacing the central difference formula with the asymmetric difference formula, i.e., the backward or forward difference formula. To guarantee the hermiticity of the Hamiltonian matrix, the backward and forward difference formula should be used alternatively according to the parity of the wave function. This provides a simple and efficient numerical prescription to solve various relativistic problems in the microscopic world.

Third order convergent finite difference method for the third order boundary value problem in ODEs

Abstract: We propose a third order convergent finite-difference method for the approximate solution of the boundary value problems. We developed our numerical technique by employing Taylor series expansion and method of undetermined coefficients. The convergence property of the proposed finite difference method discussed. To demonstrate the computational accuracy and effectiveness of the proposed method numerical results presented.

Comparative study of numerical approximation schemes for laplacian operator in polar mesh system on 9-points stencil including mixed partial derivative by finite difference method

Abstract: In this resech paper we have discussed on two different types of discretization schmes for laplacian operator which carries nine points stencil including two pair of lines symmeical and asymmetrical from planned molecule. it is well organized and modicfied five point scheme which has been devloped on finite difference method. FDM is a method which is being used for decretization of of PDES as well as ODES. In this paper we will use two different types of scheme which are devloped by FDM on polar coordinate system and in last we will discuss on error analysis, staibility and graphically behaviour of both scheme.

High-Order Compact Finite Difference Methods for Solving the High-Dimensional Helmholtz Equations

Abstract: Abstract In this paper, the sixth-order compact finite difference schemes for solving two-dimensional (2D) and three-dimensional (3D) Helmholtz equations are proposed. Firstly, the sixth-order compact difference operators for the second-order derivatives are applied to approximate the Laplace operator. Meanwhile, with the original differential equation, the sixth-order compact difference schemes are proposed. However, the truncation errors of the proposed scheme obviously depend on the unknowns, source function and wavenumber. Thus, we correct the truncation error of the sixth-order compact scheme to obtain an improved sixth-order compact scheme that is more accurate. Theoretically, the convergence and stability of the present improved method are proved. Finally, numerical tests verify that the improved schemes are more accurate.

Approximate Solution of Elliptic Poission Equation with IBVP using Finite Difference Scheme

Abstract: In this study, we have considered for numerical solution of a Poisson equation in a domain. we have presented a first and second order finite difference method for solving the system of the intial boundary value. We have presented the numerical solution of the Poisson equations in a two-dimensional finite region. Numerical Solution and exact solution for the presented method has been compared and the figures presented. The present error analysis tables demonstrate the efficiency of the method.

Hampiran Solusi Persamaan Gelombang Dua Dimensi Dengan Pendekatan Finite Difference

Abstract: The finite difference method is widely used in determining the approximate solution of a time dependent partial differential equation. The purpose of this study is to calculate the numerical solution with a finite difference method to the two dimensional wave equation. The research method used is literature study. The solution of numerical problem using the finite difference method. Discretization the two-dimensional wave equation with a central difference approach. The second step, the discretization result is simulated by Matlab software. Based on the finite difference method result, the numerical solution approximates the analytical solution of the given two dimensional wave equation. The stability requirements of numerical solution using the finite difference method is the Von Nuemann stability.

Analysis and numerical simulation of the time fractional diffusion equation by using mimetic finite differences

Abstract: This paper is devoted to the numerical treatment of time fractional diffusion equation with mixed boundary conditions. A new scheme based on the combination of the implicit finite difference method for Caputo derivative in time and the mimetic finite difference in space is considered for solving this problem. The stability analysis of the proposed scheme is given by using Von-Neumann method. The numerical results are provided to demonstrate the effectiveness of the proposed method as compared with other finite difference methods.

Analysis of Rectangular Water Tank Resting on Ground by Using Finite Difference Method

Abstract: Abstract: Water tanks are used to store water and are designed as crack free structures, to eliminate any leakage. In this research, analysis of rectangular water tank resting over ground is done by using Finite Difference Method (FDM). The finite difference method is probably the most transparent and the most general It is considered to be subjected to an arbitrary transverse uniformly distributed loading and is considered to be clamped at the two opposite edges and free at the other two edges. The ordinary Finite Difference Method (FDM) is used to solve the governing differential equation of the plate deflection. The proposed methods can be easily programmed to readily apply on a plate problem.

Two linearized finite difference schemes for time fractional nonlinear diffusion-wave equations with fourth order derivative

Abstract: In this paper, we present a finite difference and a compact finite difference schemes for the time fractional nonlinear diffusion-wave equations (TFNDWEs) with the space fourth order derivative. To reduce the smoothness requirement in time, the considered TFNDWEs are equivalently transformed into their partial integro-differential forms with the classical first order integrals and the Caputo derivative. The finite difference scheme is constructed by using Crank-Nicolson method combined with the midpoint formula, the weighted and shifted Gr$\ddot{u}$nwald difference formula and the second order convolution quadrature formula to deal with the temporal discretizations. Meanwhile, the classical central difference formula and fourth order Stephenson scheme are used in spacial direction. Then, the compact finite difference scheme is developed by using the fourth order compact difference formula for the spatial direction. The stability and convergence of the proposed schemes are strictly proved by using the discrete energy method. Finally, some numerical experiments are presented to support our theoretical results.

Optimizing Finite-Difference Operator in Seismic Wave Numerical Modeling

Abstract: The finite-difference method is widely used in seismic wave numerical simulation, imaging, and waveform inversion. In the finite-difference method, the finite difference operator is used to replace the differential operator approximately, which can be obtained by truncating the spatial convolution series. The properties of the truncated window function, such as the main and side lobes of the window function’s amplitude response, determine the accuracy of finite-difference, which subsequently affects the seismic imaging and inversion results significantly. Although numerical dispersion is inevitable in this process, it can be suppressed more effectively by using higher precision finite-difference operators. In this paper, we use the krill herd algorithm, in contrast with the standard PSO and CDPSO (a variant of PSO), to optimize the finite-difference operator. Numerical simulation results verify that the krill herd algorithm has good performance in improving the precision of the differential operator.