The Finite-Difference Scheme of Higher Order of Accuracy for the Two-Dimensional Poisson Equation in a Rectangle with Re
- PDF / 171,278 Bytes
- 12 Pages / 594 x 792 pts Page_size
- 81 Downloads / 212 Views
THE FINITE-DIFFERENCE SCHEME OF HIGHER ORDER OF ACCURACY FOR THE TWO-DIMENSIONAL POISSON EQUATION IN A RECTANGLE WITH REGARD FOR THE EFFECT OF THE DIRICHLET BOUNDARY CONDITION
UDC 519.6
N. V. Mayko
Abstracts. We investigate the finite-difference scheme of higher order of accuracy on a nine-point template for Poisson’s equation in a rectangle with the Dirichlet boundary condition. We substantiate the error estimate taking into account the influence of the boundary condition. We prove that the accuracy order is higher near the sides of the rectangle than at the inner nodes of the grid set and increase in the approximation order has no impact on the boundary effect. Keywords: Poisson’s equation, Dirichlet boundary condition, finite-difference scheme, nine-point template, difference operator, error estimate, boundary effect. INTRODUCTION In solving boundary-value and initial–boundary-value problems by a finite-difference method, it was revealed [1] that the accuracy of approximate solution is higher near the part of the boundary where the Dirichlet boundary condition is specified. The quantitative characteristics of this observation are estimates of the rate of convergence of the scheme that take into account influence of the boundary condition. Such estimates are obtained in a number of publications (for example, [2–5]). In the present paper, we will consider the standard scheme of higher order of approximation on a nine-point template for two-dimensional Poisson’s equation with the Dirichlet boundary condition on the sides of the rectangle. Important classes of applied problems (such as the problem about torsion of a prismatic rod with rectangular cross section) are known to be reduced to the solution of the Poisson equation. With the use of a difference analog of this problem and comparison theorem, a priori estimate of the accuracy of the method with regard for the boundary effect is obtained. A similar scheme is considered in [6], where the error in the norm with weight is estimated with application of another approach. STATEMENT OF THE BOUNDARY-VALUE PROBLEM AND ITS FINITE-DIFFERENCE APPROXIMATION Consider the problem
Du = - f ( x ), x Î D, u( x ) = 0, x Î G,
(1)
Taras Shevchenko National University of Kyiv, Kyiv, Ukraine, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 4, July–August, 2018, pp. 122–134. Original article submitted October 19, 2017. 624
1060-0396/18/5404-0624 ©2018 Springer Science+Business Media, LLC
where x = ( x1 , x 2 ) , D =
¶2 ¶x12
+
¶2 ¶x 22
, D = {x = ( x1 , x 2 ) : 0 < x a < la , a = 1, 2} is a rectangle, and G = ¶D is the boundary
of the rectangle D . Let us introduce grid sets
w a = { x a = ia ha , ia = 1, K , N a - 1 }, ha = la / N a , N a ³ 2 is an integer, w +a = w a U {la }, w -a = w a U {0} , w a = w a U {0} U {la }, a = 1, 2 , w = w 1 ´ w 2 , w = w1 ´ w 2 , g = w \ w and use standard notation from [7]. Let us approximate problem (1) by the difference scheme
Ly( x ) +
h12 + h22 12
L 1 L 2 y( x ) = -T1T2 f ( x ), x Î w,
(2)
y( x ) = 0, x Î g , w
Data Loading...
