Solving the Generalized Poisson Equation in Proper and Directed Interval Arithmetic
Hoffmann Tomasz 1, Marciniak A. 2,3
1 Poznan Supercomputing and Networking Center
Jana Pawla II 10, 61-139 Poznań, Poland
E-mail: tomhof@man.poznan.pl2 Institute of Computing Science, Poznan University of Technology
Piotrowo 2, 60-965 Poznań, Poland
E-mail: Andrzej.Marciniak@put.poznan.pl3 Department of Computer Science, Higher Vocational State School in Kalisz
Poznanska 201-205, 62-800 Kalisz, Poland
Received:
Received: 26 October 2016; revised: 09 December 2016; accepted: 09 December 2016; published online: 11 December 2016
DOI: 10.12921/cmst.2016.0000048
Abstract:
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.
Key words:
central-difference method, generalized Poisson equation, interval arithmetic
References:
[1] Yu. I. Shokin, Interval Analysis, Nauka, Novosibirsk, 1981.
[2] A. Marciniak, B. Szyszka, One-and Two-Stage Implicit In-
terval Methods of Runge-Kutta Type, CMST 5, 53-65 (1999).
[3] K. Gajda, A. Marciniak, B. Szyszka, Three-and Four-Stage
Implicit Interval Methods of Runge-Kutta Type, CMST 6,
41-59 (2000).
[4] A. Marciniak, Finding the Integration Interval for Inter-
val Methods of Runge-Kutta Type in Floating-Point Interval
Arithmetic, Pro Dialog, 10, 35-45 (2000).
[5] A. Marciniak, B. Szyszka, On Representation of Coefficients
in Implicit Interval Methods of Runge-Kutta Type, CMST
10(1), 57-71 (2004).
[6] A. Marciniak, Implicit Interval Methods For Solving the Ini-
tial Value Problem, Numerical Algorithms 37(1-4), 241-251
(2004).
[7] A. Marciniak, Symplectic Interval Methods for Solving
Hamiltonian Problems, Pro Dialog 22, 27-37 (2007).
[8] M. Jankowska, A. Marciniak, Implicit Interval Multistep
Methods For Solving the Initial Value Problem, CMST 8(1),
17-30 (2002).
[9] M. Jankowska, A. Marciniak, On Explicit Interval Methods
of Adams-Bashforth Type, CMST 8(2), 46-57 (2002).
[10] M. Jankowska, A. Marciniak, On Two Families Of Implicit
Interval Methods Of Adams-Moulton Type, CMST 12(2),
109-113 (2006).
[11] A. Marciniak, Multistep Interval Methods Of Nyström and
Milne-Simpson Types, CMST13(1), 23-40 (2007).
[12] A, Marciniak, On Multistep Interval Methods For Solving the
Initial Value Problem, Journal of Computational and Applied
Mathematics199(2) 229-237 (2007).
[13] K. Gajda, M. Jankowska, A. Marciniak, B. Szyszka, A Survey
Of Interval Runge-Kutta and Multistep Methods For Solving
the Initial Value Problem, in: Parallel Processing and Ap-
plied Mathematics, pages 1361-1371 Springer, 2008.
[14] A. Marciniak, Selected Interval Methods for Solving the Ini-
tial Value Problem, Publishing House of Poznań University
of Technology, 2009.
[15] R.E. Moore, The Automatic Analysis And Control Of Error In
Digital Computation Based On the Use Of Interval Numbers,
Error in Digital Computation 1, 61-130 (1965).
[16] R.E. Moore, Interval Analysis, volume 4, Prentice-Hall En-
glewood Cliffs, 1966.
[17] F. Krückeberg, Ordinary Differential Equations, Topics in
Interval Analysis, pages 91-97, (1969).
[18] N.S. Nedialkov, Interval Tools for ODEs and DAEs, In Sci-
entific Computing, Computer Arithmetic and Validated Nu-
merics, 2006. SCAN 2006, page 4, IEEE, 2006.
[19] T. Kimura M.T. Nakao, T. Kinoshita, A Priori Error
Estimates for a Full Discrete Approximation of the Heat
Equation, SIAM Journal on Numerical Analysis51(3),
1525–1541, (2013).
[20] M.T. Nakao, On Verified Computations Of Solutions for Non-
linear Parabolic Problems, Nonlinear Theory and Its Appli-
cations, IEICE 5(3), 320-338 (2014).
[21] T. Kinoshita, T. Kimura, and M.T. Nakao, On the A Poste-
riori Estimates For Inverse Operators Of Linear Parabolic
Equations With Applications to the Numerical Enclosure Of
Solutions For Nonlinear Problems, Numerische Mathematik
126(4), 679-701 (2014).
[22] A. Marciniak, An Interval Difference Method For Solving the
Poisson Equation The First Approach, Pro Dialog 24, 49-61
(2008).
[23] A. Marciniak, An Interval Version of the Crank-Nicolson
Method-The First Approach, In Applied Parallel and Scien-
tific Computing, pages 120-126 Springer, 2012.
[24] B. Szyszka, The Central Difference Interval Method for Solv-
ing the Wave Equation, In Parallel Processing and Applied
Mathematics, volume 7204 of Lecture Notes in Computer
Science, pages 523-532 Springer Berlin Heidelberg, 2012.
[25] A. Marciniak, B. Szyszka, A Central-Backward Difference
Interval Method for Solving the Wave Equation, In Applied
Parallel and Scientific Computing, pages 518-527 Springer,
2013.
[26] T. Hoffmann, A. Marciniak, B. Szyszka, Interval Versions of
Central-Difference Method for Solving the Poisson Equation
in Proper and Directed Interval Arithmetic, Foundations of
Computing and Decision Sciences 38(3), 193-206 (2013).
[27] T. Hoffmann and A. Marciniak, Solving the Poisson Equa-
tion by an Interval Difference Method of the Second Order,
CMST19(1), 13-21 (2013).
[28] Richard L Burden and J Douglas Faires, Numerical Analysis,
Brooks/Cole, 2001.
[29] David Ronald Kincaid and Elliott Ward Cheney, Numerical
Analysis: Mathematics of Scientific Computing, volume 2,
American Mathematical Soc., 2002.
[30] R. Baker Kearfott, Interval Computations: Introduction,
Uses, and Resources, Euromath Bulletin 2(1), 95-112 (1996).
[31] R.E Moore, R. Baker Kearfott, and M.J. Cloud, Introduction
to Interval Analysis, SIAM, 2009.
[32] J.R. Nagel, Solving the Generalized Poisson Equation Using
the Finite-Difference Method (FDM), Lecture Notes, Dept.
of Electrical and Computer Engineering, University of Utah,
(2011).
[33] H. Schwandt, The Solution of Nonlinear Elliptic Dirichlet
Problems on Rectangles by Almost Globally Convergent In-
terval Methods, SIAM Journal on Scientific and Statistical
Computing 6(3), 617-638 (1985).
[34] H. Schwandt, Almost Globally Convergent Interval Methods
for Discretizations of Nonlinear Elliptic Partial Differential
Equations, SIAM Journal on Numerical Analysis 23(2), 304-
324 (1986).
[35] S. Markov, On Directed Interval Arithmetic and Its Applica-
tions, in: The Journal of Universal Computer Science, pages
514-526 Springer, 1996.
[36] E.D. Popova, Extended Interval Arithmetic in IEEE Floating-
Point Environment, Interval Comput. 4, 100-129 (1994).
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.
Key words:
central-difference method, generalized Poisson equation, interval arithmetic
References:
[1] Yu. I. Shokin, Interval Analysis, Nauka, Novosibirsk, 1981.
[2] A. Marciniak, B. Szyszka, One-and Two-Stage Implicit In-
terval Methods of Runge-Kutta Type, CMST 5, 53-65 (1999).
[3] K. Gajda, A. Marciniak, B. Szyszka, Three-and Four-Stage
Implicit Interval Methods of Runge-Kutta Type, CMST 6,
41-59 (2000).
[4] A. Marciniak, Finding the Integration Interval for Inter-
val Methods of Runge-Kutta Type in Floating-Point Interval
Arithmetic, Pro Dialog, 10, 35-45 (2000).
[5] A. Marciniak, B. Szyszka, On Representation of Coefficients
in Implicit Interval Methods of Runge-Kutta Type, CMST
10(1), 57-71 (2004).
[6] A. Marciniak, Implicit Interval Methods For Solving the Ini-
tial Value Problem, Numerical Algorithms 37(1-4), 241-251
(2004).
[7] A. Marciniak, Symplectic Interval Methods for Solving
Hamiltonian Problems, Pro Dialog 22, 27-37 (2007).
[8] M. Jankowska, A. Marciniak, Implicit Interval Multistep
Methods For Solving the Initial Value Problem, CMST 8(1),
17-30 (2002).
[9] M. Jankowska, A. Marciniak, On Explicit Interval Methods
of Adams-Bashforth Type, CMST 8(2), 46-57 (2002).
[10] M. Jankowska, A. Marciniak, On Two Families Of Implicit
Interval Methods Of Adams-Moulton Type, CMST 12(2),
109-113 (2006).
[11] A. Marciniak, Multistep Interval Methods Of Nyström and
Milne-Simpson Types, CMST13(1), 23-40 (2007).
[12] A, Marciniak, On Multistep Interval Methods For Solving the
Initial Value Problem, Journal of Computational and Applied
Mathematics199(2) 229-237 (2007).
[13] K. Gajda, M. Jankowska, A. Marciniak, B. Szyszka, A Survey
Of Interval Runge-Kutta and Multistep Methods For Solving
the Initial Value Problem, in: Parallel Processing and Ap-
plied Mathematics, pages 1361-1371 Springer, 2008.
[14] A. Marciniak, Selected Interval Methods for Solving the Ini-
tial Value Problem, Publishing House of Poznań University
of Technology, 2009.
[15] R.E. Moore, The Automatic Analysis And Control Of Error In
Digital Computation Based On the Use Of Interval Numbers,
Error in Digital Computation 1, 61-130 (1965).
[16] R.E. Moore, Interval Analysis, volume 4, Prentice-Hall En-
glewood Cliffs, 1966.
[17] F. Krückeberg, Ordinary Differential Equations, Topics in
Interval Analysis, pages 91-97, (1969).
[18] N.S. Nedialkov, Interval Tools for ODEs and DAEs, In Sci-
entific Computing, Computer Arithmetic and Validated Nu-
merics, 2006. SCAN 2006, page 4, IEEE, 2006.
[19] T. Kimura M.T. Nakao, T. Kinoshita, A Priori Error
Estimates for a Full Discrete Approximation of the Heat
Equation, SIAM Journal on Numerical Analysis51(3),
1525–1541, (2013).
[20] M.T. Nakao, On Verified Computations Of Solutions for Non-
linear Parabolic Problems, Nonlinear Theory and Its Appli-
cations, IEICE 5(3), 320-338 (2014).
[21] T. Kinoshita, T. Kimura, and M.T. Nakao, On the A Poste-
riori Estimates For Inverse Operators Of Linear Parabolic
Equations With Applications to the Numerical Enclosure Of
Solutions For Nonlinear Problems, Numerische Mathematik
126(4), 679-701 (2014).
[22] A. Marciniak, An Interval Difference Method For Solving the
Poisson Equation The First Approach, Pro Dialog 24, 49-61
(2008).
[23] A. Marciniak, An Interval Version of the Crank-Nicolson
Method-The First Approach, In Applied Parallel and Scien-
tific Computing, pages 120-126 Springer, 2012.
[24] B. Szyszka, The Central Difference Interval Method for Solv-
ing the Wave Equation, In Parallel Processing and Applied
Mathematics, volume 7204 of Lecture Notes in Computer
Science, pages 523-532 Springer Berlin Heidelberg, 2012.
[25] A. Marciniak, B. Szyszka, A Central-Backward Difference
Interval Method for Solving the Wave Equation, In Applied
Parallel and Scientific Computing, pages 518-527 Springer,
2013.
[26] T. Hoffmann, A. Marciniak, B. Szyszka, Interval Versions of
Central-Difference Method for Solving the Poisson Equation
in Proper and Directed Interval Arithmetic, Foundations of
Computing and Decision Sciences 38(3), 193-206 (2013).
[27] T. Hoffmann and A. Marciniak, Solving the Poisson Equa-
tion by an Interval Difference Method of the Second Order,
CMST19(1), 13-21 (2013).
[28] Richard L Burden and J Douglas Faires, Numerical Analysis,
Brooks/Cole, 2001.
[29] David Ronald Kincaid and Elliott Ward Cheney, Numerical
Analysis: Mathematics of Scientific Computing, volume 2,
American Mathematical Soc., 2002.
[30] R. Baker Kearfott, Interval Computations: Introduction,
Uses, and Resources, Euromath Bulletin 2(1), 95-112 (1996).
[31] R.E Moore, R. Baker Kearfott, and M.J. Cloud, Introduction
to Interval Analysis, SIAM, 2009.
[32] J.R. Nagel, Solving the Generalized Poisson Equation Using
the Finite-Difference Method (FDM), Lecture Notes, Dept.
of Electrical and Computer Engineering, University of Utah,
(2011).
[33] H. Schwandt, The Solution of Nonlinear Elliptic Dirichlet
Problems on Rectangles by Almost Globally Convergent In-
terval Methods, SIAM Journal on Scientific and Statistical
Computing 6(3), 617-638 (1985).
[34] H. Schwandt, Almost Globally Convergent Interval Methods
for Discretizations of Nonlinear Elliptic Partial Differential
Equations, SIAM Journal on Numerical Analysis 23(2), 304-
324 (1986).
[35] S. Markov, On Directed Interval Arithmetic and Its Applica-
tions, in: The Journal of Universal Computer Science, pages
514-526 Springer, 1996.
[36] E.D. Popova, Extended Interval Arithmetic in IEEE Floating-
Point Environment, Interval Comput. 4, 100-129 (1994).
