A fourth-order poisson solver

International Journal for Research in Applied Science & Engineering Technology (IJRASET), 2022

In this paper, we implement the Finite Difference Method to approximate the homogeneous form of the Poisson equation. The Poisson equation is discretized using the central difference approximation of the second derivative and the grid function is determined by the five point method approximates the exact solution of the Poisson equation. The finite difference approximation is consistent and convergent. The method of solving the numerical approximation of Poisson equation is implemented using Python Programming.

AIP Advances, 2022

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)].

Discrete and Continuous Models and Applied Computational Science, 2022

The paper discusses the formulation and analysis of methods for solving the one-dimensional Poisson equation based on finite-difference approximations - an important and very useful tool for the numerical study of differential equations. In fact, this is a classical approximation method based on the expansion of the solution in a Taylor series, based on which the recent progress of theoretical and practical studies allowed increasing the accuracy, stability, and convergence of methods for solving differential equations. Some of the features of this analysis include interesting extensions to classical numerical analysis of initial and boundary value problems. In the first part, a numerical method for solving the one-dimensional Poisson equation is presented, which reduces to solving a system of linear algebraic equations (SLAE) with a banded symmetric positive definite matrix. The well-known tridiagonal matrix algorithm, also known as the Thomas algorithm, is used to solve the SLAEs....

AIP Advances, 2014

The finite difference method (FDM) based on Cartesian coordinate systems can be applied to numerical analyses over any complex domain. A complex domain is usually taken to mean that the geometry of an immersed body in a fluid is complex; here, it means simply an analytical domain of arbitrary configuration. In such an approach, we do not need to treat the outer and inner boundaries differently in numerical calculations; both are treated in the same way. Using a method that adopts algebraic polynomial interpolations in the calculation around near-wall elements, all the calculations over irregular domains reduce to those over regular domains. Discretization of the space differential in the FDM is usually derived using the Taylor series expansion; however, if we use the polynomial interpolation systematically, exceptional advantages are gained in deriving high-order differences. In using the polynomial interpolations, we can numerically solve the Poisson equation freely over any complex domain. Only a particular type of partial differential equation, Poisson's equations, is treated; however, the arguments put forward have wider generality in numerical calculations using the FDM.

The extreme high-accuracy calculation of the 1D Poisson equation by the interpolation finite difference method (IFDM) is possible. Numerical calculation errors have conventionally been evaluated by comparison with theoretical solutions. However, it is not always possible to obtain a theoretical solution. In addition, when trying to obtain numerical values from the theoretical solution, the exact numerical value may not be obtained because of inherent difficulties, that is, a theoretical solution equal to the exact numerical solution does not hold. In this paper, we focus on an ordered structure of the error calculated by the high-order accuracy finite difference (FD) scheme. This approach clarifies that in the numerical calculation of the 1D Poisson equation, which is the most basic ordinary differential equation, the error of the numerical calculation can be evaluated without comparison with the theoretical solution. Furthermore, in the numerical calculation by the IFDM, not only t...

Research Squar, Preprint, 2023

To conduct numerical calculation in the finite difference method (FDM), a calculation system should ideally be constructed to have three features: (i) the possibility of correspondence to an arbitrary boundary shape, (ii) high accuracy and (iii) high-speed calculation. In this study, the author has proposed and reported the interpolation FDM (IFDM) as a numerical calculation system with the above three characteristics. In this paper, we especially focus on (ii) high accuracy calculation. Regarding the 1D Poisson equation, the author has already reported on the overall picture of numerical calculations and proposed three schemes for high-accuracy numerical calculations: (i) the SAPI(m) scheme, (ii) SOBI(m) scheme, and (iii) CIFD(m) scheme. (m) denotes the accuracy order, which is usually an even number. Conventionally, high-order accuracy schemes up to the sixth order have been researched and reported, but theoretically, there is no upper accuracy order limit for (m) in these three s...

Journal of Computational Physics, 1997

The multigrid method is among the most efficient iterative methods to solve linear systems arising from discretiz-We combine a compact high-order difference approximation with multigrid V-cycle algorithm to solve the two-dimensional Poisson ing elliptic differential equations. It solves the error correcequation with Dirichlet boundary conditions. This scheme, along tion (coarse-grid-correction) sub-problem on the coarse with several different orderings of grid space and projection operagrids and interpolates the error correction solution back tors, is compared with the five-point formula to show the dramatic to the fine grids. Considerable computational time is saved improvement in computed accuracy, on serial and vector by doing major computational work on the coarse grids.

Journal of Computational and Applied Mathematics, 2006

New compact approximation schemes for the Laplace operator of 4th-and 6thorder are proposed. The schemes are based on a Padé approximation of the Taylor expansion for the discretized Laplace operator. The new schemes are compared with other finite difference approximations in several test cases. It is found that the new schemes are superior in performance and accuracy with respect to other methods.

Numerical Methods for Partial Differential Equations, 1998

A symbolic procedure for deriving various finite difference approximations for the three-dimensional Poisson equation is described. Based on the software package Mathematica, we utilize for the formulation local solutions of the differential equation and obtain the standard second-order scheme (7-point), three fourthorder finite difference schemes (15-point, 19-point, 21-point), and one sixth-order scheme (27-point). The symbolic method is simple and can be used to obtain the finite difference approximations for other partial differential equations.

Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 2006

We solve Poisson's equation in d = 2, 3 space dimensions by using a spectral method based on Fourier decomposition. The choice of the basis implies that Dirichlet boundary conditions on a box are satisfied. A Green's function based procedure allows us to impose Dirichlet conditions on any smooth closed boundary, by doubling the computational complexity. The error introduced by the spectral truncation and the discretization of the charge distribution is evaluated by comparison with the exact solution, known in the case of elliptical symmetry. To this end boundary conditions on an equipotential ellipse (ellipsoid) are imposed on the numerical solution. Scaling laws for the error dependence on the the number K of Fourier components for each space dimension and the number N of point charges used to simulate the charge distribution are presented and tested. A procedure to increase the accuracy of the method in the beam core region is briefly outlined.