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Strong Stability Preserving General Linear Methods

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Abstract

We investigate the strong stability preserving (SSP) general linear methods with two and three external stages and \(s\) internal stages. We also describe the construction of starting procedures for these methods. Examples of SSP methods are derived of order \(p=2, p=3\), and \(p=4\) with \(2\le s\le 10\) stages, which have larger effective Courant–Friedrichs–Levy coefficients than the class of two-step Runge–Kutta methods introduced by Jackiewicz and Tracogna, whose SSP properties were analyzes recently by Ketcheson, Gottlieb, and MacDonald, and the class of multistep multistage methods investigated by Constantinescu and Sandu. Numerical examples illustrate that the class of methods derived in this paper achieve the expected order of accuracy and do not produce spurious oscillations for discretizations of hyperbolic conservation laws, when combined with appropriate discretizations in spatial variables.

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References

  1. Albrecht, P.: Numerical treatment of O.D.E.s: the theory of \(A\)-methods. Numer. Math. 47, 59–87 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  2. Albrecht, P.: A new theoretical approach to Runge–Kutta methods. SIAM J. Numer. Anal. 24, 391–406 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  3. Albrecht, P.: Elements of a general theory of composite integration methods. Appl. Math. Comput. 31, 1–17 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  4. Albrecht, P.: The Runge–Kutta theory in a nutshell. SIAM J. Numer. Anal. 33, 1712–1735 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Albrecht, P.: The common basis of the theories of linear cyclic methods and Runge–Kutta methods. Appl. Numer. Math. 22, 3–21 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Burrage, K.: Parallel and Sequential Methods for Ordinary Differential Equations. Clarendon Press, Oxford (1995)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Butcher, J.C.: The Numerical Analysis of Ordinary Differential Equations. Wiley, New York (1987)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  11. Butcher, J.C.: Numerical Methods for Ordinary Differential Equations, 2nd edn. Wiley, Chichester (2008)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Butcher, J.C., Jackiewicz, Z.: A reliable error estimation for DIMSIMs. BIT 41, 656–665 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Butcher, J.C., Jackiewicz, Z.: A new approach to error estimation for general linear methods. Numer. Math. 95, 487–502 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Butcher, J.C., Jackiewicz, Z.: Construction of general linear methods with Runge–Kutta stability properties. Numer. Algorithms 36, 53–72 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Butcher, J.C., Jackiewicz, Z.: Uncoditionally stable general linear methods for ordinary differential equations. BIT 44, 557–570 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cardone, A., Jackiewicz, Z., Verner, J.H., Welfert, B.: Order conditions for general linear methods. Research Report 2013/01. AGH, Poland (2013), (submitted)

  18. Carpenter, M.H., Gottlieb, D., Abarbanel, S., Don, W.-S.: The theoretical accuracy of Runge-Kutta time discretizations for the initial boundary value problem: a study of the boundary error. SIAM J. Sci. Comput. 16(6), 1241–1252 (1995)

  19. Constantinescu, E.M., Sandu, A.: Optimal explicit strong-stability-preserving general linear methods. SIAM J. Sci. Comput. 32, 3130–3150 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ferracina, L., Spijker, M.N.: An extension and analysis of the Shu–Osher representation of Runge–Kutta methods. Math. Comput. 74, 201–219 (2004)

    Article  MathSciNet  Google Scholar 

  21. Ferracina, L., Spijker, M.N.: Stepsize restrictions for the total-variation-diminishing property in general Runge–Kutta methods. SIAM J. Numer. Anal. 42, 1073–1093 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ferracina, L., Spijker, M.N.: Stepsize restrictions for the total-variation-boundedness in general Runge–Kutta procedures. Appl. Numer. Math. 53, 265–279 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ferracina, L., Spijker, M.N.: Strong stability of singly-diagonally-implicit Runge–Kutta methods. Appl. Numer. Math. 58, 1675–1686 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Gottlieb, S.: On high order strong stability preserving Runge–Kutta methods and multistep time discretizations. J. Sci. Comput. 25, 105–127 (2005)

    MathSciNet  MATH  Google Scholar 

  25. Gottlieb, S., Ketcheson, D.I., Shu, Chi-Wang: High order strong stability preserving time discretizations. J. Sci. Comput. 38, 251–289 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Gottlieb, S., Ketcheson, D.I., Shu, Chi-Wang: Strong Stability Preserving Runge–Kutta and Multistep Time Discretizations. World Scientific, Hackensack (2011)

    Book  MATH  Google Scholar 

  27. Gottlieb, S., Ruuth, S.J.: Optimal strong-stability-preserving time stepping schemes with fast downwind spatial discretizations. J. Sci. Comput. 27, 289–303 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  28. Gottlieb, S., Shu, Chi-Wang, Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43, 89–112 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I, Nonstiff Problems, Second Revised Edition. Springer, Berlin (1993)

    Google Scholar 

  30. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, Second Revised Edition. Springer, Berlin (1996)

    Book  Google Scholar 

  31. Higueras, I.: On strong stability preserving time discretization methods. J. Sci. Comput. 21, 193–223 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  32. Higueras, I.: Monotonicity for Runge–Kutta methods: inner product norms. J. Sci. Comput. 24, 97–117 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  33. Higueras, I.: Representations of Runge–Kutta methods and strong stability preserving methods. SIAM J. Numer. Anal. 43, 924–948 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  34. Hundsdorfer, W., Ruuth, S.J.: On monotonicity and boundedness properties of linear multistep methods. Math. Comput. 75, 655–672 (2005)

    Article  MathSciNet  Google Scholar 

  35. Hundsdorfer, W., Ruuth, S.J., Spiteri, R.J.: Monotonicity-preserving linear multistep methods. SIAM J. Numer. Anal. 41, 605–623 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  36. Hunndsdorfer, W., Verwer, J.G.: Numerical Solution of Time-Dependent Advection–Diffusion–Reaction Equations. Springer, Berlin (2003)

    Book  Google Scholar 

  37. Izzo, G., Jackiewicz, Z.: Construction of algebraically stable DIMSIMs. J. Comput. Appl. Math. 261, 72–84 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  38. Jackiewicz, Z.: Step-control stability of diagonally implicit multistage integration methods. N. Z. J. Math. 29, 193–201 (2000)

    MathSciNet  MATH  Google Scholar 

  39. Jackiewicz, Z.: Implementation of DIMSIMs for stiff differential systems. Appl. Numer. Math. 42, 251–267 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  40. Jackiewicz, Z.: General Linear Methods for Ordinary Differential Equations. Wiley, Hoboken (2009)

    Book  MATH  Google Scholar 

  41. Jackiewicz, Z., Vermiglio, R.: General linear methods with external stages of different orders. BIT 36, 688–712 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  42. Jackiewicz, Z., Tracogna, S.: A general class of two-step Runge–Kutta methods for ordinary differential equations. SIAM. J. Numer. Anal. 32, 1390–1427 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  43. Ketcheson, D.I.: Highly efficient strong stability-preserving Runge–Kutta methods with low-storage implementations. SIAM J. Sci. Comput. 30, 2113–2136 (2008)

    Article  MathSciNet  Google Scholar 

  44. Ketcheson, D.I.: Computation of optimal monotonicity preserving general linear methods. Math. Comput. 78, 1497–1513 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  45. Ketcheson, D.I., Gottlieb, S., Macdonald, C.B.: Strong stability preserving two-step Runge–Kutta methods. SIAM J. Numer. Anal. 49, 2618–2639 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  46. Ketcheson, D.I., Macdonald, C.B., Gottlieb, S.: Optimal implicit strong stability preserving Runge–Kutta methods. Appl. Numer. Math. 59, 373–392 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  47. Koren, B.: A robust upwind discretization for advection, diffusion and source terms. In: Vreugdenhil C.B., Koren B. (eds.) Numerical Methods for Advection–Diffusion Problems. Notes Numer. Fluid Mech. 45, 117–138 (1993)

  48. Kraaijevanger, J.F.B.M.: Contractivity of Runge–Kutta methods. BIT 31, 482–528 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  49. LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  50. Ruuth, S.J., Hundsdorfer, W.: High-order linear multistep methods with general monotonicity and boundedness properties. J. Comput. Phys. 209, 226–248 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  51. Sanz-Serna, J.M., Verwer, J.G.: Convergence analysis of one-step schemes in the method of lines. Appl. Math. Comput. 31, 183–196 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  52. Sanz-Serna, J.M., Verwer, J.G., Hundsdorfer, W.H.: Convergence and order reduction of Runge–Kutta schemes applied to evolutionary problems in partial differential equations. Numer. Math. 50, 405–418 (1986)

    Article  MathSciNet  Google Scholar 

  53. Shu, Chi-Wang, Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  54. Skeel, R.: Analysis of fixed-stepsize methods. SIAM J. Numer. Anal. 13, 664–685 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  55. Spijker, M.N.: Stepsize conditions for general monotonicity in numerical initial value problems. SIAM J. Numer. Anal. 45, 1226–1245 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  56. Spiteri, R.J., Ruuth, S.J.: A new class of optimal high-order strong-stability-preserving time discretization methods. SIAM J. Numer. Anal. 40, 469–491 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  57. Wright, W.: General linear methods with inherent Runge–Kutta stability. Ph.D. thesis, The University of Auckland, New Zealand (2002)

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Acknowledgments

The results reported in this paper were obtained during the visit of the first author (GI) to the Arizona State University in the Spring semester of 2014. This author wish to express his gratitude to the School of Mathematical and Statistical Sciences for hospitality during this visit. The authors would also like to express their gratitude to anonymous referees for their useful comments which helped to improve presentation of this paper.

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Authors and Affiliations

  1. Dipartimento di Matematica e Applicazioni, Università di Napoli-“Federico II”, 80126, Napoli, Italy

    Giuseppe Izzo

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

    Zdzislaw Jackiewicz

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

    Zdzislaw Jackiewicz

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  1. Giuseppe Izzo
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  2. Zdzislaw Jackiewicz
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Correspondence to Giuseppe Izzo.

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The work of the first author was partially supported by GNCS-INdAM.

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Izzo, G., Jackiewicz, Z. Strong Stability Preserving General Linear Methods. J Sci Comput 65, 271–298 (2015). https://doi.org/10.1007/s10915-014-9961-7

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