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Construction of the Nordsieck second derivative methods with RK stability for stiff ODEs

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Abstract

In this paper, we study the construction and implementation of special Nordsieck second derivative general linear methods of order p and stage order \(q=p\) in which the number of input and output values is \(r=p\) rather than \(r=p+1\). We will construct A- and L-stable methods of orders three and four in this form with Runge–Kutta stability properties. The efficiency of the constructed methods and reliability of the proposed error estimates are shown by implementing of the methods in a variable stepsize environment on some well-known stiff problems.

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References

  • Abdi A (2016) Construction of high-order quadratically stable second-derivative general linear methods for the numerical integration of stiff ODEs. J Comput Appl Math 303:218–228

    Article  MathSciNet  MATH  Google Scholar 

  • Abdi A, Braś M, Hojjati G (2014) On the construction of second derivative diagonally implicit multistage integration methods for ODEs. Appl Numer Math 76:1–18

    Article  MathSciNet  MATH  Google Scholar 

  • Abdi A, Butcher JC (2012) Order bounds for second derivative approximations. BIT Numer Math 52:273–281

    Article  MathSciNet  MATH  Google Scholar 

  • Abdi A, Hojjati G (2011) An extension of general linear methods. Numer Algor 57(2):149–167

    Article  MathSciNet  MATH  Google Scholar 

  • Abdi A, Hojjati G (2015) Implementation of Nordsieck second derivative methods for stiff ODEs. Appl Numer Math 94:241–253

    Article  MathSciNet  MATH  Google Scholar 

  • Abdi A, Hojjati G (2011) Maximal order for second derivative general linear methods with Runge-Kutta stability. Appl Numer Math 61(10):1046–1058

    Article  MathSciNet  MATH  Google Scholar 

  • Braś M, Cardone A, D’Ambrosio R (2013) Implementation of explicit Nordsieck methods with inherent quadratic stability. Math Model Anal 18(2):289–307

    Article  MathSciNet  MATH  Google Scholar 

  • Butcher JC (2008) Numerical methods for ordinary differential equations. Wiley, NewYork

    Book  MATH  Google Scholar 

  • Butcher JC (1996) On the convergence of numerical solutions to ordinary differential equations. Math Comp 20:1–10

    Article  MathSciNet  MATH  Google Scholar 

  • Butcher JC (1993) Diagonally-implicit multistage integration methods. Appl Numer Math 11:347–363

    Article  MathSciNet  MATH  Google Scholar 

  • Butcher JC (2006) General linear methods. Acta Numer 15:157–256

    Article  MathSciNet  MATH  Google Scholar 

  • Butcher JC (2016) Numerical methods for ordinary differential equations. Wiley, New York

    Book  MATH  Google Scholar 

  • Butcher JC, Hojjati G (2005) Second derivative methods with RK stability. Numer Algor 40(4):415–429

    Article  MathSciNet  MATH  Google Scholar 

  • Butcher JC, Jackiewicz Z (1996) Construction of diagonally implicit general linear methods type 1 and 2 for ordinary differential equations. Appl Numer Math 21:385–415

    Article  MathSciNet  MATH  Google Scholar 

  • Butcher JC, Jackiewicz Z (1998) Construction of high order diagonally implicit multistage integration methods for ordinary differential equations. Appl Numer Math. 27:1–12

    Article  MathSciNet  MATH  Google Scholar 

  • Butcher JC, Jackiewicz Z (1993) Diagonally implicit general linear methods for ordinary differential equations. BIT 33(3):452–472

    Article  MathSciNet  MATH  Google Scholar 

  • Butcher JC, Jackiewicz Z (1997) Implementation of diagonally implicit multistage integration methods for ordinary differential equations. SIAM J Numer Anal 34(6):2119–2141

    Article  MathSciNet  MATH  Google Scholar 

  • Butcher JC, Wright WM (2003) A transformation relating explicit and diagonally-implicit general linear methods. Appl Numer Math 44:313–327

    Article  MathSciNet  MATH  Google Scholar 

  • Butcher JC, Wright WM (2003) The construction of practical general linear methods. BIT Numer Math 43:695–721

    Article  MathSciNet  MATH  Google Scholar 

  • Cash JR (1981) Second derivative extended backward differentiation formulas for the numerical integration of stiff systems. SIAM J Numer Anal 18:21–36

    Article  MathSciNet  MATH  Google Scholar 

  • Chan RPK, Tsai AYJ (2010) On explicit two-derivative Runge–Kutta methods. Numer Algorithms 53:171194

    Article  MathSciNet  Google Scholar 

  • Conte D, D’Ambrosio R, Jackiewicz Z (2010) Two-step Runge–Kutta methods with quadratic stability functions. J Sci Comput 44:191–218

    Article  MathSciNet  MATH  Google Scholar 

  • D’Ambrosio R, Esposito E, Paternoster B (2012) General linear methods for \(y^{\prime \prime }=f(y(t))\). Numer Algorihms 61:331–349

    Article  MATH  Google Scholar 

  • D’Ambrosio R, De Martino G, Paternoster B (2016) General Nyström methods in Nordsieck form: error analysis. J Comput Appl Math 292:694–702

    Article  MathSciNet  MATH  Google Scholar 

  • D’Ambrosio R, Paternoster B (2014) P-stable general Nyström methods for \(y^{\prime \prime }=f(y(t))\). J Comput Appl Math 262:271–280

    Article  MathSciNet  MATH  Google Scholar 

  • Enright WH (1974) Second derivative multistep methods for stiff ordinary differential equations. SIAM J Numer Anal 11:321–331

    Article  MathSciNet  MATH  Google Scholar 

  • Ezzeddine AK, Hojjati G, Abdi A (2014) Sequential second derivative general linear methods for stiff systems. Bull Iran Math Soc 40(1):83–100

    MathSciNet  MATH  Google Scholar 

  • Ezzeddine AK, Hojjati G, Abdi A (2015) Perturbed second derivative multistep methods. J Numer Math 23:235–245

    Article  MathSciNet  MATH  Google Scholar 

  • Hairer E, Wanner G (1996) Solving ordinary differential equations II: stiff and differential-algebraic problems. Springer, Berlin

    Book  MATH  Google Scholar 

  • Hojjati G, Ardabili MY Rahimi, Hosseini SM (2006) New second derivative multistep methods for stiff systems. Appl Math Model 30:466–476

    Article  MATH  Google Scholar 

  • Huang SJY (2005) Implementation of general linear methods for stiff ordinary differential equations. Phd thesis, Department of mathematics Auckland University

  • Jackiewicz Z (2009) General linear methods for ordinary differential equations. Wiley, New Jersey

    Book  MATH  Google Scholar 

  • Lee JHJ (2004) Numerical methods for ordinary differential equations: a survey of some standard methods. MSc thesis, Department of Mathematic Auckland University

  • Movahedinejad A, Hojjati G, Abdi A (2016) Second derivative general linear methods with inherent Runge–Kutta stability. Numer Algor 73:371–389

    Article  MathSciNet  MATH  Google Scholar 

  • Movahedinejad A, Hojjati G, Abdi A (2017) Construction of Nordsieck second derivative general linear methods with inherent quadratic stability. Math Model Anal 22:60–77

    Article  MathSciNet  MATH  Google Scholar 

  • Prothero A, Robinson A (1974) On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations. Math Comp 28:145–162

    Article  MathSciNet  MATH  Google Scholar 

  • Wright WM (2002) Expilicit general linear methods with inherent Runge–Kutta stability. Numer Algor 31:381–399

    Article  MATH  Google Scholar 

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Author notes

  1. B. Behzad

    Present address: Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran

Authors and Affiliations

  1. Department of Mathematics, Lorestan University, Khorramabad, Iran

    B. Behzad & B. Ghazanfari

  2. Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran

    A. Abdi

Authors
  1. B. Behzad
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  2. B. Ghazanfari
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  3. A. Abdi
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Correspondence to A. Abdi.

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Communicated by Frederic Valentin.

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Behzad, B., Ghazanfari, B. & Abdi, A. Construction of the Nordsieck second derivative methods with RK stability for stiff ODEs. Comp. Appl. Math. 37, 5098–5112 (2018). https://doi.org/10.1007/s40314-018-0619-1

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  • DOI: https://doi.org/10.1007/s40314-018-0619-1

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