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Adapted BDF Algorithms: Higher-order Methods and Their Stability

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We present BDF type formulas of high-order (4, 5 and 6), capable of the exact integration (with only round-off errors) of differential equations whose solutions are linear combinations of an exponential with parameter A and ordinary polynomials. For A = 0, the new formulas reduce to the classical BDF formulas. Theorems of the local truncation error reveal the good behavior of the new methods with stiff problems. Plots of their 0-stability regions in terms of the eigenvalues of the parameter A h are provided. Plots of their absolute stability regions that include the whole of the negative real axis are provided. The weights of the method usually require the evaluation of a matrix exponential. However, if the dimension of the matrix is large, we shall not perform this calculus and shall only approximate those coefficients once. Numerical examples underscore the efficiency of the proposed codes, especially when one is integrating stiff oscillatory problems.

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References

  1. Butcher J.C., Chen D.J.L. (2001). On the implementation of ESIRK methods for stiff IVPs. Numer. Algorithms 26: 201–218

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  3. Cash J.R. (1980). On the integration of stiff systems of ODE’s using extended backward differentiation formulae. Numer. Math. 37: 235–246

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  6. Cox S.M., Matthews P.C. (2002). Exponential time differencing for stiff systems. J. Comput. Phys. 176: 430–455

    Article  MATH  Google Scholar 

  7. Enright W.H. (1974). Optimal second derivative methods for stiff systems. In: Willoughby R.A. (eds) Stiff Differential Systems. Plenum Press, New York, pp. 95–111

    Google Scholar 

  8. Eriksson K., Johnson C., Logg A. (2003). Explicit time-stepping for stiff ODE’s. SIAM J. Sci. Comput. 25(4): 1142–1157

    Article  MATH  Google Scholar 

  9. Hairer, E., Norsett, P., and Wanner, G. (1993). Solving Ordinary Differential Equations I, Springer, Berlin. Second Revised Edition.

  10. Kaps, P. (1981). Rosenbrock-type methods, In Dahlquist, G., and Jeltsch, R. (ed.), Numerical Methods for Stiff Initial Value Problems, Bericht nr. 9. Inst für Geometrie und Praktische Mathematik der RWTH Aachen, Templergraben 55, D-5100 Aachen.

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

  12. Hochbruck M., Lubich Ch., Selhofer H. (1998). Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput. 19: 1552–1574

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  16. Kassam A.K., Trefethen Ll.N. (2005). Fourth-order time stepping for stiff PDEs. SIAM J. Sci. Comput. 26: 1214–1233

    Article  MATH  Google Scholar 

  17. Lambert, J. D. (1991). Numerical Methods for Ordinary Differential Systems. The initial Value Problem, Wiley, Chichester.

  18. Moler C.B., Van Loan C.F. (1978). Nineteen dubious ways to compute the exponential of a matrix. SIAM Rev. 29(4): 801–836

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  20. Robertson, H. H. (1966). The Solution of a Set Of Reaction Rate equations, Academic New York, pp. 178–182.

  21. Sidge R.B. (1998). Expokit: software package for computing matrix exponentials. ACM Trans. Math. Softw. 24(1): 130–156

    Article  Google Scholar 

  22. Van de Vyver H. (2005). Frequency evaluation for exponentially fitted Runge–Kutta methods. J. Comput. Appl. Math. 184(2): 442–463

    Article  MATH  Google Scholar 

  23. Vigo-Aguiar J., Ferrándiz J.M. (1998). A general procedure for the adaptation of multistep algorithms to the integration of oscillatory problems. SIAM J. Numer. Anal. 35(4): 1684–1708

    Article  MATH  Google Scholar 

  24. Vigo-Aguiar J. (1999). An approach to variable coefficients multistep methods for special differential equations. Int. J. Appl. Math. 1(8): 911–921

    Google Scholar 

  25. Vigo-Aguiar J., Martín-Vaquero J., Criado R. (2005). On the stability of exponential fitting BDF algorithms. J. Comput. Appl. Math. 175(1): 183–194

    Article  MATH  Google Scholar 

  26. Walz, G. (1988). Numerical Computation of the Matrix Exponential, Constructive theory of functions, Publ. House Bulgar. Acad. Sci., MathSciNet, Sofia, pp. 478–481.

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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. Adapted BDF Algorithms: Higher-order Methods and Their Stability. J Sci Comput 32, 287–313 (2007). https://doi.org/10.1007/s10915-007-9132-1

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