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Erschienen in: Numerical Algorithms 4/2021

14.10.2020 | Original Paper

Second derivative backward differentiation formulae for ODEs based on barycentric rational interpolants

verfasst von: Ali Abdi, Gholamreza Hojjati

Erschienen in: Numerical Algorithms | Ausgabe 4/2021

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Abstract

For their several attractive features from the viewpoint of the numerical computations, linear barycentric rational interpolants have been recently used to construct various numerical methods for solving different classes of equations. In this paper, we introduce a family of linear multistep second derivative methods together with a starting procedure based on barycentric rational interpolants. The order of convergence and linear stability properties of the proposed methods have been investigated. To validate the theoretical results and efficiency of the methods, some numerical experiments have been provided.

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Literatur
1.
Zurück zum Zitat Abdi, A., Behzad, B.: Efficient Nordsieck second derivative general linear methods: construction and implementation. Calcolo 55(28), 1–16 (2018) MathSciNetMATH Abdi, A., Behzad, B.: Efficient Nordsieck second derivative general linear methods: construction and implementation. Calcolo 55(28), 1–16 (2018) MathSciNetMATH
2.
Zurück zum Zitat Abdi, A., Berrut, J.P., Hosseini, S.A.: Explicit methods basedon barycentric rational interpolants for solving non-stiff Volterra integral equations. submitted Abdi, A., Berrut, J.P., Hosseini, S.A.: Explicit methods basedon barycentric rational interpolants for solving non-stiff Volterra integral equations. submitted
3.
Zurück zum Zitat Abdi, A., Berrut, J.P., Hosseini, S.A.: The linear barycentric rational method for a class of delay Volterra integro-differential equations. J. Sci. Comput. 75, 1757–1775 (2018) MathSciNetMATHCrossRef Abdi, A., Berrut, J.P., Hosseini, S.A.: The linear barycentric rational method for a class of delay Volterra integro-differential equations. J. Sci. Comput. 75, 1757–1775 (2018) MathSciNetMATHCrossRef
4.
Zurück zum Zitat Abdi, A., Braś, M., Hojjati, G.: On the construction of second derivative diagonally implicit multistage integration methods for ODEs. Appl. Numer. Math. 76, 1–18 (2014) MathSciNetMATHCrossRef Abdi, A., Braś, M., Hojjati, G.: On the construction of second derivative diagonally implicit multistage integration methods for ODEs. Appl. Numer. Math. 76, 1–18 (2014) MathSciNetMATHCrossRef
6.
Zurück zum Zitat Abdi, A., Hojjati, G.: Implementation of Nordsieck second derivative methods for stiff ODEs. Appl. Numer. Math. 94, 241–253 (2015) MathSciNetMATHCrossRef Abdi, A., Hojjati, G.: Implementation of Nordsieck second derivative methods for stiff ODEs. Appl. Numer. Math. 94, 241–253 (2015) MathSciNetMATHCrossRef
7.
Zurück zum Zitat Abdi, A., Hosseini, S.A.: The barycentric rational difference-quadrature scheme for systems of Volterra integro-differential equations. SIAM J. Sci. Comput. 40, A1936–A1960 (2018) MathSciNetMATHCrossRef Abdi, A., Hosseini, S.A.: The barycentric rational difference-quadrature scheme for systems of Volterra integro-differential equations. SIAM J. Sci. Comput. 40, A1936–A1960 (2018) MathSciNetMATHCrossRef
8.
Zurück zum Zitat Abdi, A., Hosseini, S.A., Podhaisky, H.: Adaptive linear barycentric rational finite differences method for stiff ODEs. J. Comput. Appl. Math. 357, 204–214 (2019) MathSciNetMATHCrossRef Abdi, A., Hosseini, S.A., Podhaisky, H.: Adaptive linear barycentric rational finite differences method for stiff ODEs. J. Comput. Appl. Math. 357, 204–214 (2019) MathSciNetMATHCrossRef
9.
Zurück zum Zitat Abdi, A., Hosseini, S.A., Podhaisky, H.: Numerical methods based on the Floater–Hormann interpolants for stiff VIEs. Numer. Algor. to appear Abdi, A., Hosseini, S.A., Podhaisky, H.: Numerical methods based on the Floater–Hormann interpolants for stiff VIEs. Numer. Algor. to appear
10.
Zurück zum Zitat Berrut, J.P.: Rational functions for guaranteed and experimentally well-conditioned global interpolation. Comput. Math. Appl. 15, 1–16 (1988) MathSciNetMATHCrossRef Berrut, J.P.: Rational functions for guaranteed and experimentally well-conditioned global interpolation. Comput. Math. Appl. 15, 1–16 (1988) MathSciNetMATHCrossRef
11.
Zurück zum Zitat Berrut, J.P., Floater, M.S., Klein, G.: Convergence rates of derivatives of a family of barycentric rational interpolants. Appl. Numer. Math. 61, 989–1000 (2011) MathSciNetMATHCrossRef Berrut, J.P., Floater, M.S., Klein, G.: Convergence rates of derivatives of a family of barycentric rational interpolants. Appl. Numer. Math. 61, 989–1000 (2011) MathSciNetMATHCrossRef
12.
Zurück zum Zitat Berrut, J.P., Hosseini, S.A., Klein, G.: The linear barycentric rational quadrature method for Volterra integral equations. SIAM J. Sci. Comput. 36, A105–A123 (2014) MathSciNetMATHCrossRef Berrut, J.P., Hosseini, S.A., Klein, G.: The linear barycentric rational quadrature method for Volterra integral equations. SIAM J. Sci. Comput. 36, A105–A123 (2014) MathSciNetMATHCrossRef
13.
14.
Zurück zum Zitat Butcher, J.C.: Numerical Methods for Ordinary Differential Equations. Wiley, Chichester (2016) MATHCrossRef Butcher, J.C.: Numerical Methods for Ordinary Differential Equations. Wiley, Chichester (2016) MATHCrossRef
16.
Zurück zum Zitat Cash, J.R.: On the integration of stiff systems of ODEs using extended backward differentiation formulae. Numer. Math. 34, 235–246 (1980) MathSciNetMATHCrossRef Cash, J.R.: On the integration of stiff systems of ODEs using extended backward differentiation formulae. Numer. Math. 34, 235–246 (1980) MathSciNetMATHCrossRef
17.
Zurück zum Zitat Cash, J.R.: Second derivative extended backward differentiation formulas for the numerical integration of stiff systems. SIAM J. Numer. Anal. 18, 21–36 (1981) MathSciNetMATHCrossRef Cash, J.R.: Second derivative extended backward differentiation formulas for the numerical integration of stiff systems. SIAM J. Numer. Anal. 18, 21–36 (1981) MathSciNetMATHCrossRef
18.
Zurück zum Zitat Cash, J.R.: The integration of stiff initial value problems in ODEs using modified extended backward differentiation formulae. Comput. Math. Appl. 9, 645–657 (1983) MathSciNetMATHCrossRef Cash, J.R.: The integration of stiff initial value problems in ODEs using modified extended backward differentiation formulae. Comput. Math. Appl. 9, 645–657 (1983) MathSciNetMATHCrossRef
20.
Zurück zum Zitat Dahlquist, G.: Convergence and stability in the numerical integration of ordinary differential equations. Math. Scand. 4, 33–53 (1956) MathSciNetMATHCrossRef Dahlquist, G.: Convergence and stability in the numerical integration of ordinary differential equations. Math. Scand. 4, 33–53 (1956) MathSciNetMATHCrossRef
21.
Zurück zum Zitat Enright, W.H.: Second derivative multistep methods for stiff ordinary differential equations. SIAM J. Numer. Anal. 11, 321–331 (1974) MathSciNetMATHCrossRef Enright, W.H.: Second derivative multistep methods for stiff ordinary differential equations. SIAM J. Numer. Anal. 11, 321–331 (1974) MathSciNetMATHCrossRef
22.
Zurück zum Zitat Floater, M.S., Hormann, K.: Barycentric rational interpolation with no poles and high rates of approximation. Numer. Math. 107, 315–331 (2007) MathSciNetMATHCrossRef Floater, M.S., Hormann, K.: Barycentric rational interpolation with no poles and high rates of approximation. Numer. Math. 107, 315–331 (2007) MathSciNetMATHCrossRef
23.
Zurück zum Zitat Fredebeul, C.: A–BDF: a generalization of the backward differentiation formulae. SIAM J. Numer. Anal. 35, 1917–1938 (1998) MathSciNetMATHCrossRef Fredebeul, C.: A–BDF: a generalization of the backward differentiation formulae. SIAM J. Numer. Anal. 35, 1917–1938 (1998) MathSciNetMATHCrossRef
24.
Zurück zum Zitat Gear, C.W.: Numerical Initial Value Problems in Ordinary Differential Equations. Prentice-hall, Englewood Cliffs, (1971) MATH Gear, C.W.: Numerical Initial Value Problems in Ordinary Differential Equations. Prentice-hall, Englewood Cliffs, (1971) MATH
25.
Zurück zum Zitat Hairer, E., Wanner G.: Solving Ordinary Differential Equations II: Stiff and Differential–Algebraic Problems. Springer, Berlin (2010) MATH Hairer, E., Wanner G.: Solving Ordinary Differential Equations II: Stiff and Differential–Algebraic Problems. Springer, Berlin (2010) MATH
26.
Zurück zum Zitat Hojjati, G., Rahimi Ardabili, M.Y., Hosseini, S.M.: A–EBDF: an adaptive method for numerical solution of stiff systems of ODEs. Math. Comput. Simul. 66, 33–41 (2004) MathSciNetMATHCrossRef Hojjati, G., Rahimi Ardabili, M.Y., Hosseini, S.M.: A–EBDF: an adaptive method for numerical solution of stiff systems of ODEs. Math. Comput. Simul. 66, 33–41 (2004) MathSciNetMATHCrossRef
27.
Zurück zum Zitat Hojjati, G., Rahimi Ardabili, M.Y., Hosseini, S.M.: New second derivative multistep methods for stiff systems. Appl. Math. Model. 30, 466–476 (2006) MATHCrossRef Hojjati, G., Rahimi Ardabili, M.Y., Hosseini, S.M.: New second derivative multistep methods for stiff systems. Appl. Math. Model. 30, 466–476 (2006) MATHCrossRef
28.
Zurück zum Zitat Hosseini, S.M., Hojjati, G.: Matrix free MEBDF method for the solution of stiff systems of ODEs. Math. Comput. Model. 29, 67–77 (1999) MathSciNetMATHCrossRef Hosseini, S.M., Hojjati, G.: Matrix free MEBDF method for the solution of stiff systems of ODEs. Math. Comput. Model. 29, 67–77 (1999) MathSciNetMATHCrossRef
29.
Zurück zum Zitat Jackiewicz, Z.: General Linear Methods for Ordinary Differential Equations. John Wiley, New Jersey (2009) MATHCrossRef Jackiewicz, Z.: General Linear Methods for Ordinary Differential Equations. John Wiley, New Jersey (2009) MATHCrossRef
30.
Zurück zum Zitat Klein, G.: Applications of linear barycentric rational interpolation. PhD thesis, University of Fribourg (2012) Klein, G.: Applications of linear barycentric rational interpolation. PhD thesis, University of Fribourg (2012)
31.
Zurück zum Zitat Klein, G., Berrut, J.P.: Linear rational finite differences from derivatives of barycentric rational interpolants. SIAM J. Numer. Anal. 50, 643–656 (2012) MathSciNetMATHCrossRef Klein, G., Berrut, J.P.: Linear rational finite differences from derivatives of barycentric rational interpolants. SIAM J. Numer. Anal. 50, 643–656 (2012) MathSciNetMATHCrossRef
Metadaten
Titel
Second derivative backward differentiation formulae for ODEs based on barycentric rational interpolants
verfasst von
Ali Abdi
Gholamreza Hojjati
Publikationsdatum
14.10.2020
Verlag
Springer US
Erschienen in
Numerical Algorithms / Ausgabe 4/2021
Print ISSN: 1017-1398
Elektronische ISSN: 1572-9265
DOI
https://doi.org/10.1007/s11075-020-01020-6

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