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Extrapolation-based implicit-explicit general linear methods

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Abstract

For many systems of differential equations modeling problems in science and engineering, there are natural splittings of the right hand side into two parts, one non-stiff or mildly stiff, and the other one stiff. For such systems implicit-explicit (IMEX) integration combines an explicit scheme for the non-stiff part with an implicit scheme for the stiff part. In a recent series of papers two of the authors (Sandu and Zhang) have developed IMEX GLMs, a family of implicit-explicit schemes based on general linear methods. It has been shown that, due to their high stage order, IMEX GLMs require no additional coupling order conditions, and are not marred by order reduction. This work develops a new extrapolation-based approach to construct practical IMEX GLM pairs of high order. We look for methods with large absolute stability region, assuming that the implicit part of the method is A- or L-stable. We provide examples of IMEX GLMs with optimal stability properties. Their application to a two dimensional test problem confirms the theoretical findings.

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References

  1. Ascher, U.M., Ruuth, S.J., Spiteri, R.J.: Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. Appl. Numer. Math. 25, 151–167 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ascher, U.M., Ruuth, S.J., Wetton, B.TR.: Implicit-explicit methods for time-dependent partial differential equations. SIAM J. Numer. Anal. 32(3), 797–823 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  3. Braś, M., Cardone, A.: Construction of efficient general linear methods for non-stiff differential systems. Math. Model. Anal. 17(2), 171–189 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. Burrage, K.: Parallel and Sequential Methods for Ordinary Differential Equations. Clarendon Press, New York (1995)

    MATH  Google Scholar 

  6. Burrage, K., Butcher, J.C.: Non-linear stability of a general class of differential equation methods. BIT 20, 185–203 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  7. Butcher JC: The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods.Wiley-Interscience, New York (1987)

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  9. Butcher, J.C.: Diagonally-implicit multi-stage integration methods. Appl. Numer. Math. 11(5), 347–363 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  10. Butcher, J.C.: Numerical Methods for Ordinary Differential Equations. Wiley, Chichester (2003)

    Book  MATH  Google Scholar 

  11. Butcher, J.C.: General linear methods. Acta Numer. 15, 157–256 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Calvo, M.P., de Frutos, J., Novo, J.: Linearly implicit Runge-Kutta methods for advection-reaction-diffusion equations. Appl. Numer. Math. 37(4), 535–549 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  13. Cardone, A., Jackiewicz, Z., Mittelmann, H.D.: Optimization-based search for Nordsieck methods of high order with quadratic stability. Math. Model. Anal. 17(3), 293–308 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  14. Cardone, A., Jackiewicz, Z., Sandu, A., Zhang H: Extrapolation-Based Implicit-Explicit General Linear Methods for Ordinary Differential Equations (2013). arXiv:1304.2276

  15. Cardone, A., Jackiewicz, Z.: Explicit Nordsieck methods with quadratic stability. Numer. Algoritm 60, 1–25 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  16. Crouzeix, M.: Une méthode multipas implicite-explicite pour l’approximation des équations d’évolution paraboliques. Numer. Math. 35(3), 257–276 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  17. Frank, J., Hundsdorfer, W., Verwer, J.G.: On the stability of implicit-explicit linear multistep methods. Appl. Numer. Math. 25(2–3), 193–205 (1997). Special issue on time integration (Amsterdam, 1996)

    Article  MATH  MathSciNet  Google Scholar 

  18. Hairer, E., Nørsett, S.P., Wanner, G.: Solving ordinary differential equations. I, Springer Series in Computational Mathematics, vol. 8, 2nd ed. Springer-Verlag, Berlin (1993). Nonstiff problems

    Google Scholar 

  19. Hairer, E., Wanner, G.: Solving ordinary differential equations. II, Springer Series in Computational Mathematics, vol. 14. Springer-Verlag, Berlin (2010). Stiff and differential-algebraic problems, Second revised edition, paperback

    Google Scholar 

  20. Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer-Verlag, New York (2002)

    Book  Google Scholar 

  21. Hundsdorfer, W., Ruuth, S.J.: IMEX extensions of linear multistep methods with general monotonicity and boundedness properties. J. Comput. Phys. 225, 2016–2042 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  22. Hundsdorfer, W., Verwer, J.: Numerical solution of time-dependent advection-diffusion-reaction equations. Springer Series in Computational Mathematics, vol. 33. Springer-Verlag, Berlin (2003)

    Book  Google Scholar 

  23. Jackiewicz, Z.: General linear methods for ordinary differential equations. Wiley, Hoboken (2009)

    Book  MATH  Google Scholar 

  24. Kennedy, C.A., Carpenter, M.H.: Additive Runge-Kutta schemes for convection-diffusion-reaction equations. Appl. Numer. Math. 44(1–2), 139–181 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  25. Liska, R., Wendroff, B.: Composite schemes for conservation laws. SIAM J. Numer. Anal. 35(6), 2250–2271 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  26. Pareschi, L., Russo, G.: Implicit-explicit Runge-Kutta schemes for stiff systems of differential equations. In: Recent Trends in Numerical Analysis, Advances Theory Computational Mathematics, vol. 3, pp. 269-288. Nova Scientific Publications, Huntington (2001)

  27. Pareschi, L., Giovanni, R.: Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation. J. Sci. Comput. 25(1–2), 129–155 (2005)

    MATH  MathSciNet  Google Scholar 

  28. Prothero, A., Robinson, A.: On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations. Math. Comput. 28(125), 145–162 (1974)

    Article  MathSciNet  Google Scholar 

  29. Varah, J.M.: Stability restrictions on second order, three level finite difference schemes for parabolic equations. SIAM J. Numer. Anal. 17(2), 300–309 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  30. Wright, W.M.: The construction of order 4 DIMSIMs for ordinary differential equations. Numer. Algoritm 26(2), 123–130 (2001)

    Article  MATH  Google Scholar 

  31. Zhang, H., Sandu, A.: A second-order diagonally-implicit-explicit multi-stage integration method. Procedia CS 9, 1039–1046 (2012)

    Google Scholar 

  32. Zhang, H., Sandu, A.: Partitioned and Implicit-explicit General Linear Methods For Ordinary Differential Equations (2013). arXiv:1302.2689

  33. Zharovski, E., Sandu, A.: A class of Implicit-explicit Two-Step Runge-Kutta Methods. Technical report TR-12-08, Computer Science, Virginia Tech (2012). http://eprints.cs.vt.edu/archive/00001183

  34. Zhong, X.: Additive semi-implicit Runge-Kutta methods for computing high-speed nonequilibrium reactive flows. J. Comput. Phys. 128(1), 19–31 (1996)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Salerno, Fisciano, (Sa), 84084, Italy

    Angelamaria Cardone

  2. Department of Mathematics, Arizona State University, Tempe, AZ, 85287, USA

    Zdzislaw Jackiewicz

  3. AGH University of Science and Technology, Kraków, Poland

    Zdzislaw Jackiewicz

  4. Department of Computer Science, Virginia Polytechnic Institute & State University, Blacksburg, VA, 24061, USA

    Adrian Sandu & Hong Zhang

Authors
  1. Angelamaria Cardone
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  2. Zdzislaw Jackiewicz
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  3. Adrian Sandu
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  4. Hong Zhang
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Correspondence to Angelamaria Cardone.

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Cardone, A., Jackiewicz, Z., Sandu, A. et al. Extrapolation-based implicit-explicit general linear methods. Numer Algor 65, 377–399 (2014). https://doi.org/10.1007/s11075-013-9759-y

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  • DOI: https://doi.org/10.1007/s11075-013-9759-y

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