Interval Versions of Central-Difference Method for Solving the Poisson Equation in Proper and Directed Interval Arithmetic

Abstract

To study the Poisson equation, the central-difference method is often used. This method has the local truncation error of order O(h2 +k2), where h and k are mesh constants. Using this method in conventional floating-point arithmetic, we get solutions including the method, representation and rounding errors. Therefore, we propose interval versions of the central-difference method in proper and directed interval arithmetic. Applying such methods in floating-point interval arithmetic allows one to obtain solutions including all possible numerical errors. We present numerical examples from which it follows that the presented interval method in directed interval arithmetic is a little bit better than the one in proper interval arithmetic, i.e. the intervals of solutions are smaller. It appears that applying both proper and directed interval arithmetic the exact solutions belong to the interval solutions obtained.

Nakao’s method and an interval difference scheme of second order for solving the elliptic BVP

“…For the problem (1) with c = 0, interval difference methods of second order based on proper and directed interval arithmetic we have developed in details in [11][12][13][14][15]. 2 In this section we expand our second order method in proper interval arithmetic for the case c = c (x, y).…”

“…Note that in our previous papers [11][12][13][14][15] on the right-hand side of (19) we wrote f ij instead of −f ij . The modification in this paper is a consequence of Nakao's notation of elliptic boundary value problem used in [6].…”

“…respectively, where C is the thinnest machine interval representing c. These equations are also useful for solving in interval arithmetic the Poisson equation, in which c = 0 (see [11][12][13][14][15]). Let us recall that Nakao's method is not applicable to the Poisson equation (see Sec.…”

“…In our previous paper we have considered interval difference methods of second and four order for solving the Poisson equation with Dirichlet's conditions in interval proper and direct floating-point arithmetic [11][12][13][14][15]. In this paper we generalize our second order method to solve the problems considered by Nakao in [6].…”

An interval version of Cauchy’s problem for the wave equation

“…The interval versions of difference methods for elliptic equations in [3] and [4] and for parabolic equations in [5] are considered. The study of interval difference method for hyperbolic equation in floating-point interval arithmetic is continued by author.…”