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Exponential fitted Gauss, Radau and Lobatto methods of low order

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Abstract

Several exponential fitting Runge-Kutta methods of collocation type are derived as a generalization of the Gauss, Radau and Lobatto traditional methods of two steps. The new methods are capable of the exact integration (with only round-off errors) of differential equations whose solutions are linear combinations of an exponential and ordinary polynomials. Theorems of the truncation error reveal the good behavior of the new methods for stiff problems. Plots of their absolute stability regions that include the whole of the negative real axis are provided. A different procedure to find the parameter of the method is proposed. The variable step Radau method of two stages is derived. Finally, numerical examples underscore the efficiency of the proposed codes, especially when they are integrating stiff problems.

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References

  1. Curtiss, C.F., Hirschfelder, J.O.: Integration of stiff equations. Proc. Natl. Acad. Sci. U. S. A. 38, 235–243 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aitken, R.C.: Stiff Computation. Oxford University Press, New York (1985)

    Google Scholar 

  3. Spijker, M.N.: Stiffness in the numerical initial-value problems. J. Comput. Appl. Math. 72, 393–406 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Springer, Berlin (1993)

    Google Scholar 

  5. Higham, D.J.: Stiffness of ODEs. BIT 33, 285–303 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  6. Lambert, J.D.: Numerical Methods for Ordinary Differential Systems. The Initial Value Problem. Wiley, Chichester (1991)

    MATH  Google Scholar 

  7. Gear, C.W.: Algorithm 407-DIFSUB for solution of ordinary differential equations. Commun. ACM 14, 185–190 (1971)

    Article  MathSciNet  Google Scholar 

  8. Hindmarsh, A.C.: LSODE and LSODI, two new initial value ordinary differential equation solvers. ACM Signum Newslett. 15, 10–11 (1980)

    Article  Google Scholar 

  9. Brown, P.N., Byrne, G.D., Hidmarsh, A.C.: VODE: a variable-coefficient ODE solver. SIAM J. Sci. Statist. Comput. 10, 1038–1051 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  10. Petzold, L.R.: A description of DASSL: a differential-algebraic system solver. In: Stepleman, R.S., et al. (eds.) IMACS Trans. Scientific Computing, pp. 65–68. North-Holland, Amsterdam (1993)

    Google Scholar 

  11. Cash, J.R.: The integration of stiff initial value problems in ODE’s using modified extended backward differentiation formulae. Comput. Math. Appl. 9, 645–657 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fredebul, C.: A-BDF: a generalization of the backward differentiation formulae. SIAM J. Numer. Anal. 35, 1917–1938 (1998)

    Article  MathSciNet  Google Scholar 

  13. Ixaru, L.Gr., Rizea, M., Vanden Berghe, G., De Meyer, H.: Weights of the exponential fitting multistep algorithms for ODEs. J. Comput. Appl. Math. 132, 83–93 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ixaru, L.Gr., Vanden Berghe, G., De Meyer, H.: Frequency evaluation in exponential fitting multistep algorithms for ODEs. J. Comput. Appl. Math. 140, 423–434 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ixaru, L.Gr., Vanden Berghe, G., De Meyer, H.: Exponentially fitted variable two-step BDF algorithms for first order ODEs. Comput. Phys. Commun. 100, 56–70 (2003)

    Article  MathSciNet  Google Scholar 

  16. Martin-Vaquero, J., Vigo-Aguiar, J.: Adapted BDF Algorithms: higher-order methods and their stability. J. Sci. Comput. 32, 287–313 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Alexander, R.: Diagonally implicit Runge-Kutta methods for stiff ODEs. SIAM J. Numer. Anal. 14, 1006–1021 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  18. Nørsett, S.P.: Semi explicit Runge-Kutta methods. Mathematics and Computing Rpt. N. 6/74, University of Trodheim (1974)

  19. Darvishi, M.T., Khani, F., Kheybari, S.: Spectral collocation solution of a generalized Hirota-Satsuma coupled KdV equation. Int. J. Comput. Math. 84, 541–552 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Khater, A.H., Temsah, R.S.: Numerical solutions of some nonlinear evaluation equations by Chebyshev spectral collocation methods. Int. J. Comput. Math. 84, 305–316 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Simos, T.E., Vigo-Aguiar, J.: A modified Runge-Kutta method with phase-lag of order infinity for the numerical solution of the Schrodinger equation and related problems. Comput. Chem. 25, 275–281 (2001)

    Article  MATH  Google Scholar 

  22. Simos, T.E., Vigo-Aguiar, J.: Symmetric eighth algebraic order methods with minimal phase-lag for the numerical solution of the Schrodinger equation. J. Math. Chem. 31, 135-144 (2002)

    Article  MathSciNet  Google Scholar 

  23. Liniger, W., Willoughby, R.: Efficient integration for stiff systems of ordinary differential equations. SIAM J. Numer. Anal. 7, 47–66 (1970)

    Article  MathSciNet  Google Scholar 

  24. Van de Vyver, H.: Frequency evaluation for exponentially fitted Runge-Kutta methods. J. Comput. Appl. Math. 184, 442–463 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Vanden Berghe, G., Van Daale, M., Vande Vyver, H.: Exponential fitted Runge-Kutta methods of collocation type: fixed or variable knot points. J. Comput. Appl. Math. 159, 217–239 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  26. Butcher, J.C.: The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods. Wiley, Chichester (1987)

    MATH  Google Scholar 

  27. Vigo-Aguiar, J., Martín-Vaquero, J., Ramos, H.: Exponential fitting BDF-Runge-Kutta algorithms. Comput. Phys. Commun. 178, 15–34 (2008)

    Article  Google Scholar 

  28. Coleman, J.P., Ixaru, L.Gr.: P-stability and exponential-fitting methods for y”=f(x,y). IMA J. Numer. Anal. 16, 179–199 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  29. Scherer, R.: A necessary condition for B-stability. BIT 19, 111–115 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  30. Stetter, H.J.: Analysis of Discretization Methods for Ordinary Differential Equations. Springer, Berlin (1973)

    MATH  Google Scholar 

  31. Martín-Vaquero, J., Vigo-Aguiar, J.: Exponential fitting BDF algorithms: explicit and implicit 0-stable methods. J. Comput. Appl. Math. 192, 100–113 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. Vigo-Aguiar, J., Martín-Vaquero, J., Criado, R.: On the stability of exponential fitting BDF algorithms. J. Comput. Appl. Math. 175, 183–194 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  33. Vigo-Aguiar, J., Martín-Vaquero, J.: Exponential fitting BDF algorithms and their properties. Appl. Math. Comput. 190, 80–110 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  34. Edwards, W.S., Tuckerman, L.S., Friesner, R.A., Sorensen, D.C.: Krylov methods for the incompressible Navier-Stokes equations. J. Comput. Phys. 110, 82–102 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  35. Friesner, R.A., Tuckerman, L.S., Dornblaser, B.C., Russo, T.C.: A method for exponential propagation of large systems of stiff nonlinear differential equations. J. Sci. Comput. 4, 327–354 (1989)

    Article  MathSciNet  Google Scholar 

  36. Gallopoulos, E., Saad, Y.: Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Stat. Comput. 13, 1236–1264 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  37. Kosloff, R.: Propagation methods for quantum molecular dynamics. Annu. Rev. Phys. Chem. 45, 145–178 (1994)

    Article  Google Scholar 

  38. Sidje, R.B.: Expokit: software package for computing matrix exponentials. ACM Trans. Math. Softw. 24, 130–156 (1998)

    Article  MATH  Google Scholar 

  39. Frank, J.E., van der Houwen, P.J.: Parallel Iteration of the Extended Backward Differentiation Formulas. Report MAS-R9913, CWI (1999)

  40. Kaps, P.: Rosenbrock-type methods. In: Dahlquist, G., Jeltsch, R. (eds.) Numerical Methods for Stiff Initial Value Problems. Bericht nr. 9, Inst für Geometrie und Praktische Mathematik der RWTH Aachen, Aachen (1981)

  41. Butcher, J.C., Rattenbury, N.: ARK methods for stiff problems. Appl. Numer. Math. 53, 165–181 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. ETS Ingenieros industriales, Universidad de Salamanca, Bejar, Spain

    J. Martín-Vaquero

  2. Facultad de Ciencias, Universidad de Salamanca, Salamanca, Spain

    J. Vigo-Aguiar

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  1. J. Martín-Vaquero
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  2. J. Vigo-Aguiar
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Correspondence to J. Martín-Vaquero.

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Martín-Vaquero, J., Vigo-Aguiar, J. Exponential fitted Gauss, Radau and Lobatto methods of low order. Numer Algor 48, 327–346 (2008). https://doi.org/10.1007/s11075-008-9202-y

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