Finite-difference methods for solving 1D Poisson problem

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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. The second part presents a solution method based on an analytical representation of the exact inverse matrix of a discretized version of the Poisson equation. Expressions for inverse matrices essentially depend on the types of boundary conditions in the original setting. Variants of inverse matrices for the Poisson equation with different boundary conditions at the ends of the interval under study are presented - the Dirichlet conditions at both ends of the interval, the Dirichlet conditions at one of the ends and Neumann conditions at the other. In all three cases, the coefficients of the inverse matrices are easily found and the algorithm for solving the problem is practically reduced to multiplying the matrix by the vector of the right-hand side.

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1. Introduction Applied mathematical models are mainly based on the use of partial differential equations [1]. The solution must satisfy a given equation of mathematical physics and some additional relations, which are, first, boundary and initial conditions. The most important for applications [2] are secondorder equations - elliptic, parabolic, and hyperbolic. Currently for equations of mathematical physics, methods of numerical solution and the appropriate software [3], [4], as well as computer algebra systems (CASs) such as Sage, Mathematica, Maxima and Maple are actively developed to implement these methods. Many features of stationary problems of mathematical physics described by elliptic equations of the second order can be illustrated by considering the simplest boundary value problems for an ordinary differential equation of the second order. Perhaps the simplest second-order elliptic equation is the Poisson equation. Let us consider some methods for the numerical solution of this equation and compare the investigated methods. The Poisson equation [1] is a special case of the heat conduction equation describing the dependence of the temperature of a medium on spatial coordinates and time, and the heat capacity and thermal conductivity of the medium (in the general case, inhomogeneous) are considered to be given. We will consider the problem of finding the steady-state distribution of density or temperature (e.g., when the distribution of sources does not depend on time). In this case, terms with time derivatives are eliminated from the non-stationary equation and a stationary heat equation is obtained, which belongs to the class of elliptic equations. A two-point boundary value problem is the problem of finding a solution to an ordinary differential equation or second-order systems in the interval

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About the authors

Serge Ndayisenga

Peoples’ Friendship University of Russia (RUDN University)

Author for correspondence.
Email: 1032195775@rudn.ru
ORCID iD: 0000-0002-9297-9839

Student of Department of Applied Probability and Informatics

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation

Leonid A. Sevastianov

Peoples’ Friendship University of Russia (RUDN University); Bogoliubov Laboratory of Theoretical Physics Joint Institute for Nuclear Research

Email: sevastianov-la@rudn.ru
ORCID iD: 0000-0002-1856-4643

Doctor of Physical and Mathematical Sciences, Professor of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University), Leading Researcher of Bogoliubov Laboratory of Theoretical Physics, JINR

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation; 6, Joliot-Curie St., Dubna, Moscow Region, 141980, Russian Federation

Konstantin P. Lovetskiy

Peoples’ Friendship University of Russia (RUDN University)

Email: lovetskiy-kp@rudn.ru
ORCID iD: 0000-0002-3645-1060

Candidate of Physical and Mathematical Sciences, Associate Professor of Department of Applied Probability and Informatics

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation

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Copyright (c) 2022 Ndayisenga S., Sevastianov L.A., Lovetskiy K.P.

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