Geodesic
Solving the order reduction phenomenon in variable step size quasi-consistent Nordsieck methods
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 11, pp. 2004-2022
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The phenomenon is studied of reducing the order of convergence by one in some classes of variable step size Nordsieck formulas as applied to the solution of the initial value problem for a first-order ordinary differential equation. This phenomenon is caused by the fact that the convergence of fixed step size Nordsieck methods requires weaker quasi-consistency than classical Runge–Kutta formulas, which require consistency up to a certain order. In other words, quasi-consistent Nordsieck methods on fixed step size meshes have a higher order of convergence than on variable step size ones. This fact creates certain difficulties in the automatic error control of these methods. It is shown how quasi-consistent methods can be modified so that the high order of convergence is preserved on variable step size meshes. The regular techniques proposed can be applied to any quasi-consistent Nordsieck methods. Specifically, it is shown how this technique performs for Nordsieck methods based on the multistep Adams–Moulton formulas, which are the most popular quasi-consistent methods. The theoretical conclusions of this paper are confirmed by the numerical results obtained for a test problem with a known solution. 
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G. Yu. Kulikov. Solving the order reduction phenomenon in variable step size quasi-consistent Nordsieck methods. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 11, pp. 2004-2022. http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_11_a7/

[1] Kalitkin H. H., Chislennye metody, Nauka, M., 1978 | MR

[2] Bakhvalov N. S., Zhidkov N. P., Kobelkov G. M., Chislennye metody, Nauka, M., 1987 | MR | Zbl

[3] Khairer E., Nersett S., Vanner G., Reshenie obyknovennykh differentsialnykh uravnenii. Nezhestkie zadachi, Mir, M., 1990 | MR

[4] Dekker K., Verver Ya., Ustoichivost metodov Runge–Kutty dlya zhestkikh nelineinykh differentsialnykh uravnenii, Mir, M., 1988 | MR

[5] Shalashilin V. I., Kuznetsov E. B., Metod prodolzheniya resheniya po parametru i nailuchshaya parametrizatsiya v prikladnoi matematike i mekhanike, Editorial URSS, M., 1999 | MR

[6] Butcher J. C., Numerical methods for ordinary differential equations, John Wiley and Sons, Chichester, 2008 | MR | Zbl

[7] Nordsieck A., “On the numerical integration of ordinary differential equations”, Math. Comp., 16 (1962), 22–49 | DOI | MR | Zbl

[8] Skeel R. D., “Equivalent forms for multistep formulas”, Math. Comput., 33 (1979), 1229–1250 | DOI | MR | Zbl

[9] Byrne G. D., Hindmarsh A. C., “A polyalgorithm for the numerical solution of ordinary differential equations”, ACM Trans. on Math. Software, 1 (1975), 71–96 | DOI | MR | Zbl

[10] Hindmarsh A. C., Byrne G. D., An experimental package for the integration of systems of ordinary differential equations, Report UCID-30112, Lowerence Livermore Laboratory, 1975

[11] Gear C. W., “Algorithm 407, DIFSUB for solution of ordinary differential equations”, Comm. ACM, 14 (1971), 185–190 | DOI | MR

[12] Hindmarsh A. C., GEAR: Ordinary differential equation system solver, Report UCID-30001. Revision 3, Lowerence Livermore Laboratory, 1974

[13] Hindmarsh A. C., “LSODE and LSOSI, two new initial value ordinary differential equation solvers”, ACM SIGNUM Newsletter, 15 (1980), 10–11 | DOI

[14] Butcher J. C., Chartier P., Jackiewicz Z., “Nordsieck representation of DIMSIMs”, Numer. Algorithms, 16 (1997), 209–230 | DOI | MR | Zbl

[15] Butcher J. C., Jackiewicz Z., “Implementation of diagonally implicit multistage integration methods for ordinary differential equations”, SIAM J. Numer. Anal., 34 (1997), 2119–2141 | DOI | MR | Zbl

[16] Podhaisky H., Weiner R., Schmitt B. A., “Linearly-implicit two-step methods and their implementation in Nordsieck form”, Appl. Numer. Math., 56 (2006), 374–387 | DOI | MR | Zbl

[17] Kulikov G. Yu., Shindin S. K., “Local and global error estimation in Nordsieck methods”, Russ. J. Numer. Anal. Math. Model., 23 (2008), 567–595 | DOI | MR | Zbl

[18] Kulikov G. Yu., “Global error control in adaptive Nordsieck methods”, SIAM J. Sci. Comput., 34 (2012), A839–A860 | DOI | MR | Zbl

[19] Skeel R. D., “Analysis of fixed-stepsize methods”, SIAM J. Numer. Anal., 13 (1976), 664–685 | DOI | MR | Zbl

[20] Skeel R. D., Jackson L. W., “Consistency of Nordsieck methods”, SIAM J. Numer. Anal., 14 (1977), 910–924 | DOI | MR | Zbl

[21] Kulikov G. Yu., “On quasi-consistent integration by Nordsieck methods”, J. Comput. Appl. Math., 225 (2009), 268–287 | DOI | MR | Zbl

[22] Gray W. G., Muccino J. C., “Accuracy of multivalue methods during change of time-step size: Adams–Moulton”, Numer. Meth. Partial Diff. Equations, 16 (2000), 312–326 | 3.0.CO;2-D class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[23] Kulikov G. Yu., Shindin S. K., “Nordsieck methods on nonuniform grids: Stability and order reduction phenomenon”, Math. Comput. Simul., 72 (2006), 47–56 | DOI | MR | Zbl

[24] Gear C. W., Tu W., “The effects of variable mesh size on the stability of multistep methods”, SIAM J. Numer. Anal., 11 (1974), 1025–1043 | DOI | MR | Zbl

[25] Skeel R. D., Jackson L. W., “The stability of variable-stepsize Nordsieck methods”, SIAM J. Numer. Anal., 20 (1983), 840–853 | DOI | MR | Zbl

[26] Calvo M., Lisbona F., Montijano J., “On the stability of variable-stepsize Nordsieck BDF methods”, SIAM J. Numer. Anal., 24 (1987), 844–854 | DOI | MR | Zbl

[27] Calvo M., Lisbona F., Montijano J., “On the stability of interpolatory variable-stepsize Adams methods in Nordsieck form”, SIAM J. Numer. Anal., 26 (1989), 946–962 | DOI | MR | Zbl

[28] Kulikov G. Yu., Shindin S. K., “Ob odnom sposobe upravleniya globalnoi oshibkoi mnogoshagovykh metodov”, Zh. vychisl. matem. i matem. fiz., 40:9 (2000), 1308–1329 | MR | Zbl

[29] Kulikov G. Yu., Shindin S. K., “O mnogoshagovykh metodakh interpolyatsionnogo tipa s avtomaticheskim kontrolem globalnoi oshibki”, Zh. vychisl. matem. i matem. fiz., 44:8 (2004), 1388–1409 | MR | Zbl

[30] Kulikov G. Yu., Shindin S. K., “Global error estimation and extrapolated multistep methods for index 1 differential-algebraic systems”, BIT, 45 (2005), 517–542 | DOI | MR | Zbl

[31] Kulikov G. Yu., Shindin S. K., “Ob effektivnom vychislenii asimptoticheski vernykh otsenok lokalnoi i globalnoi oshibok dlya mnogoshagovykh metodov s postoyannymi koeffitsientami”, Zh. vychisl. matem. i matem. fiz., 44:5 (2004), 840–861 | MR

[32] Bellman R., Vvedenie v teoriyu matrits, Nauka, M., 1969 | MR | Zbl

[33] Abramowitz M., Stegun I. A., Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications, New York, 1992 | MR