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

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Hoffmann T. , Marciniak A.

Computational Methods in Science and Technology

2013

tom Vol. 19, No. 1

13--21

The paper deals with an interval difference method for solving the Poisson equation based on the conventional central-difference method. We present the interval method in full details. The method is constructed in such a way that the exact solution is included in the interval solution obtained. Some numerical results obtained in ﬂoating-point interval arithmetic are also presented

Solving the Generalized Poisson Equation in Proper and Directed Interval Arithmetic

100%

Hoffmann T. , Marciniak A.

Computational Methods in Science and Technology

2016

tom Vol. 22, No. 4

225--232

In the paper some interval methods for solving the generalized Poisson equation (GPE) are presented. The main aim of this work is focused on providing such algorithms for solving this type of equation that are able to store information about potentially made numerical errors inside the results. In order to cope with these assumptions the floating-point interval arithmetic is used. We proposed to use interval versions of the central-difference method for two types of interval arithmetic: proper and directed. In the experimental part of this paper both arithmetics for three examples of GPE are compared

Interval versions of central-difference method for solving the poisson equation in proper and directed interval arithmetic

75%

Hoffmann T. , Marciniak A. , Szyszka B.

Foundations of Computing and Decision Sciences

2013

tom Vol. 38, No. 3

193--206

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.