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BIT Numerical Mathematics

, Volume 55, Issue 4, pp 1219–1241 | Cite as

Faster SDC convergence on non-equidistant grids by DIRK sweeps

  • Martin WeiserEmail author
Article

Abstract

Spectral deferred correction methods for solving stiff ODEs are known to converge reasonably fast towards the collocation limit solution on equidistant grids, but show a less favourable contraction on non-equidistant grids such as Radau-IIa points. We interprete SDC methods as fixed point iterations for the collocation system and propose new DIRK-type sweeps for stiff problems based on purely linear algebraic considerations. Good convergence is recovered also on non-equidistant grids. The properties of different variants are explored on a couple of numerical examples.

Keywords

Spectral deferred correction Diagonally implicit Runge–Kutta Contraction rate 

Mathematics Subject Classification

65L06 65M20 65M70 

Notes

Acknowledgments

The author is indebted to Bodo Erdmann for many fruitful discussions and thorough computational assistance. Partial funding by the DFG Research Center Matheon “Mathematics for key technologies”, projects A17 and F9, is gratefully acknowledged.

References

  1. 1.
    Auzinger, W., Hofstätter, H., Koch, O., Kreuzer, W., Weinmüller, E.: Superconvergent defect correction algorithms. WSEAS Trans. Syst. 4, 1378–1383 (2004)zbMATHGoogle Scholar
  2. 2.
    Auzinger, W., Hofstätter, H., Kreuzer, W., Weinmüller, E.: Modified defect correction algorithms for ODEs. Part I: general theory. Numer. Algor. 36(2), 135–156 (2004)CrossRefzbMATHGoogle Scholar
  3. 3.
    Auzinger, W., Hofstätter, H., Kreuzer, W., Weinmüller, E.: Modified defect correction algorithms for ODEs. Part II: stiff initial value problems. Numer. Algor. 40(3), 285–303 (2005)CrossRefzbMATHGoogle Scholar
  4. 4.
    Bowen, M.M.: A spectral deferred correction method for solving cardiac models. PhD thesis, Duke University, Durham (2011)Google Scholar
  5. 5.
    Bu, S., Huang, J., Minion, M.L.: Semi-implicit Krylov deferred correction methods for differential algebraic equations. Math. Comput. 81, 2127–2157 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Christlieb, A.J., Macdonald, C.B., Ong, B.W.: Parallel high-order integrators. SIAM J. Sci. Comput. 32(2), 818–835 (2010)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Christlieb, A.J., Ong, B.W., Qiu, J.-M.: Comments on high order integrators embedded within integral deferred correction methods. Commun. Appl. Math. Comput. Sci 4(1), 27–56 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Deuflhard, P., Weiser, M.: Adaptive numerical solution of PDEs. de Gruyter, Germany (2012)Google Scholar
  9. 9.
    Dutt, A., Greengard, L., Rokhlin, V.: Spectral deferred correction methods for ordinary differential equations. BIT 40(2), 241–266 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Emmett, M., Minion, M.L.: Toward an efficient parallel in time method for partial differential equations. Commun. App. Math. Comp. Sci. 7(1), 105–132 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Frank, R., Überhuber, C.W.: Iterated defect correction for the efficient solution of stiff systems of ordinary differential equations. BIT 17, 146–159 (1977)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Gustafsson, B., Kress, W.: Deferred correction methods for initial value problems. BIT 41(5), 986–995 (2001)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Hansen, A.C., Strain, J.: On the order of deferred correction. Appl. Num. Math. 61, 961–973 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Huang, J., Jia, J., Minion, M.L.: Accelerating the convergence of spectral deferred correction methods. J. Comp. Phys. 214(2), 633–656 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Huang, J., Jia, J., Minion, M.L.: Arbitrary order Krylov deferred correction methods for differential algebraic equations. J. Comp. Phys. 221(2), 739–760 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Kaps, P., Rentrop, P.: Generalized Runge–Kutta methods of order four with stepsize control for stiff ordinary differential equations. Numer. Math. 33, 55–68 (1979)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Layton, A.T., Minion, M.L.: Implications of the choice of quadrature nodes for Picard integral deferred corrections methods for ordinary differential equations. BIT 45, 341–373 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Layton, A.T., Minion, M.L.: Implications of the choice of predictors for semi-implicit Picard integral deferred correction methods. Commun. App. Math. Comp. Sci. 2(1), 1–34 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Ostermann, A., Roche, M.: Rosenbrock methods for partial differential equations and fractional order of convergence. SIAM J. Numer. Anal. 30(4), 1084–1098 (1993)CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Prothero, A., Robinson, A.: On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations. Math. Comput. 28, 145–162 (1974)CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    Schild, K.H.: Gaussian collocation via defect correction. Numer. Math. 58, 369–386 (1990)CrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    Tang, T., Xie, H., Yin, X.: High-order convergence of spectral deferred correction methods on general quadrature nodes. J. Sci. Comput. 1–13 (2012)Google Scholar
  23. 23.
    Zadunaisky, P.E.: On the estimation of errors propagated in the numerical integration of ODEs. Numer. Math. 27, 21–39 (1976)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Zuse Institute BerlinTakustr. 7BerlinGermany

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