Solving the Poisson Equation by an Interval Difference Method of the Second Order

Hoffmann, Tomasz; Marciniak, Andrzej

doi:10.12921/cmst.2013.19.01.13-21

Cited by 9 publications

(8 citation statements)

References 3 publications

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“…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).…”

Section: An Interval Difference Scheme Of Second Ordermentioning

confidence: 99%

“…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].…”

Section: An Interval Difference Scheme Of Second Ordermentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

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

Marciniak¹

2019

CMST

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In the paper we compare Nakao's method to our interval difference scheme of second order. Repeating some computational examples of Nakao, we have observed that our implementation of his method gives better results. Moreover, it appears that the presented interval difference scheme gives better enclosures of exact solutions than Nakao's method. Wę also point out that the considered interval method can be used to solve the Poisson equation with Dirichlet's condition, for which Nakao's method is not applicable.

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Section: An Interval Difference Scheme Of Second Ordermentioning

confidence: 99%

Section: An Interval Difference Scheme Of Second Ordermentioning

confidence: 99%

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

Marciniak¹

2019

CMST

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“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Szyszka¹

2015

AIP Conference Proceedings

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The paper is devoted to interval versions of Cauchy's problem in homogeneous wave equation. The initial condition approximated by the Taylor series of the various orders (as well as with the approximations of the corresponding derivatives) are considered. Interval versions of conventional methods are constructed so as, they contain truncation errors. Truncation errors have been estimated for each interval variant of Cauchy's condition. All variants were used in solving the wave equation by the central difference interval method. Additionally, all computation have been implemented in floating point interval arithmetic to obtain the solutions, in the form of intervals, which include round-off errors.

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“…In our previous papers [3], [4] we have considered an interval difference method for solving the Poisson equation (1)…”

Section: Introductionmentioning

confidence: 99%

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

Hoffmann¹,

Marciniak²,

Szyszka³

2013

Foundations of Computing and Decision Sciences

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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.

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Solving the Poisson Equation by an Interval Difference Method of the Second Order

Cited by 9 publications

References 3 publications

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

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

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

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

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