Applications of Mathematics, Vol. 62, No. 2, pp. 135-169, 2017


Time discretizations for evolution problems

Miloslav Vlasák

Received September 26, 2016.  First published March 6, 2017.

Abstract:  The aim of this work is to give an introductory survey on time discretizations for liner parabolic problems. The theory of stability for stiff ordinary differential equations is explained on this problem and applied to Runge-Kutta and multi-step discretizations. Moreover, a natural connection between Galerkin time discretizations and Runge-Kutta methods together with order reduction phenomenon is discussed.
Keywords:  time discretizations; parabolic PDEs; stiff ODEs; Runge-Kutta methods; multi-step methods
Classification MSC:  65J10, 65L04, 65L20
DOI:  10.21136/AM.2017.0268-16


References:
[1] G. Akrivis, C. Makridakis, R. H. Nochetto: Galerkin and Runge-Kutta methods: unified formulation, a posteriori error estimates and nodal superconvergence. Numer. Math. 118 (2011), 429-456. DOI 10.1007/s00211-011-0363-6 | MR 2810802 | Zbl 1228.65125
[2] R. Alexander: Diagonally implicit Runge-Kutta methods for stiff O.D.E.'s. SIAM J. Numer. Anal. 14 (1977), 1006-1021. DOI 10.1137/0714068 | MR 0458890 | Zbl 0374.65038
[3] D. Boffi: Finite element approximation of eigenvalue problems. Acta Numer. 19 (2010), 1-120. DOI 10.1017/S0962492910000012 | MR 2652780 | Zbl 1242.65110
[4] P. Brenner, M. Crouzeix, V. Thomée: Single-step methods for inhomogeneous linear differential equations in Banach space. RAIRO, Anal. Numér. 16 (1982), 5-26. DOI 10.1051/m2an/1982160100051 | MR 0648742 | Zbl 0477.65040
[5] J. C. Butcher: Implicit Runge-Kutta processes. Math. Comput. 18 (1964), 50-64. DOI 10.2307/2003405 | MR 0159424 | Zbl 0123.11701
[6] E. A. Coddington, N. Levinson: Theory of Ordinary Differential Equations. McGraw-Hill Book Company, New York (1955). MR 0069338 | Zbl 0064.33002
[7] M. Crouzeix, W. H. Hundsdorfer, M. N. Spijker: On the existence of solutions to the algebraic equations in implicit Runge-Kutta methods. BIT 23 (1983), 84-91. DOI 10.1007/BF01937328 | MR 0689606 | Zbl 0506.65030
[8] M. Crouzeix, F. J. Lisbona: The convergence of variable-stepsize, variable-formula, multistep methods. SIAM J. Numer. Anal. 21 (1984), 512-534. DOI 10.1137/0721037 | MR 0744171 | Zbl 0542.65038
[9] M. Crouzeix, P.-A. Raviart: Approximation des équations d'évolution linéaires par des méthodes à pas multiples. C. R. Acad. Sci., Paris, Sér. A 283 (1976), 367-370. MR 0426434 | Zbl 0361.65064
[10] C. F. Curtiss, J. O. Hirschfelder: Integration of stiff equations. Proc. Natl. Acad. Sci. USA 38 (1952), 235-243. DOI 10.1073/pnas.38.3.235 | MR 0047404 | Zbl 0046.13602
[11] G. Dahlquist: Convergence and stability in the numerical integration of ordinary differential equations. Math. Scand. 4 (1956), 33-53. DOI 10.7146/math.scand.a-10454 | MR 0080998 | Zbl 0071.11803
[12] K. Dekker: On the iteration error in algebraically stable Runge-Kutta methods. Report NW 138/82, Math. Centrum, Amsterdam (1982).
[13] V. Dolejší, M. Feistauer: Discontinuous Galerkin Method. Analysis and Applications to Compressible Flow. Springer Series in Computational Mathematics 48, Springer, Cham (2015). DOI 10.1007/978-3-319-19267-3 | MR 3363720 | Zbl 06467550
[14] B. L. Ehle: On Padé approximations to the exponential function and A-stable methods for the numerical solution of initial value problems. Thesis (Ph.D)-University of Waterloo, Ontario (1969). MR 2716012
[15] R. Frank, J. Schneid, C. W. Ueberhuber: Order results for implicit Runge-Kutta methods applied to stiff systems. SIAM J. Numer. Anal. 22 (1985), 515-534. DOI 10.1137/0722031 | MR 0787574 | Zbl 0577.65056
[16] C. W. Gear: Numerical Initial Value Problems in Ordinary Differential Equations. Prentice-Hall, Englewood Cliffs (1971). MR 0315898 | Zbl 1145.65316
[17] C. W. Gear, K. W. Tu: The effect of variable mesh size on the stability of multistep methods. SIAM J. Numer. Anal. 11 (1974), 1025-1043. DOI 10.1137/0711079 | MR 0368436 | Zbl 0292.65041
[18] R. D. Grigorieff: Stability of multistep-methods on variable grids. Numer. Math. 42 (1983), 359-377. DOI 10.1007/BF01389580 | MR 0723632 | Zbl 0554.65051
[19] R. D. Grigorieff, H. J. Pfeiffer: Numerik gewöhnlicher Differentialgleichungen. Band 2: Mehrschrittverfahren. Teubner Studienbücher: Mathematik. B. G. Teubner, Stuttgart (1977). DOI 10.1007/978-3-322-91202-2 | MR 0657222 | Zbl 0372.65025
[20] A. Guillou, J. L. Soulé: La résolution numérique des problèmes différentiels aux conditions initiales par des méthodes de collocation. Rev. Franç. Inform. Rech. Opér. 3 (1969), 17-44. DOI 10.1051/m2an/196903r300171 | MR 0280008 | Zbl 0214.15005
[21] E. Hairer, S. P. Nørsett, G. Wanner: Solving Ordinary Differential Equations. I: Nonstiff Problems. Springer Series in Computational Mathematics 8, Springer, Berlin (1993). DOI 10.1007/978-3-540-78862-1 | MR 1227985 | Zbl 0789.65048
[22] E. Hairer, G. Wanner: Solving Ordinary Differential Equations. II: Stiff and Differential-Algebraic Problems. Springer Series in Computational Mathematics 14, Springer, Berlin (1996). DOI 10.1007/978-3-642-05221-7 | MR 1439506 | Zbl 0859.65067
[23] P. Henrici: Discrete Variable Methods in Ordinary Differential Equations. John Wiley and Sons, New York (1962). MR 0135729 | Zbl 0112.34901
[24] M. Hochbruck, A. Ostermann: Exponential integrators. Acta Numerica 19 (2010), 209-286. DOI 10.1017/S0962492910000048 | MR 2652783 | Zbl 1242.65109
[25] B. L. Hulme: One-step piecewise polynomial Galerkin methods for initial value problems. Math. Comput. 26 (1972), 415-426. DOI 10.2307/2005168 | MR 0321301 | Zbl 0265.65038
[26] J. Kuntzmann: Neuere Entwicklungen der Methode von Runge und Kutta. Z. Angew. Math. Mech. 41 (1961), T28-T31. DOI 10.1002/zamm.19610411317 | Zbl 0106.10403
[27] C. Lubich: On the convergence of multistep methods for nonlinear stiff differential equations. Numer. Math. 58 (1991), 839-853. DOI 10.1007/BF01385657 | MR 1098868 | Zbl 0729.65055
[28] H. Padé: Sur la représentation approchée d'une fonction par des fractions rationnelles. Ann. Sci. Éc. Norm. Supér. (3) 9 (1892), 3-93. MR 1508880 | JFM 24.0360.02
[29] A. Prothero, A. Robinson: On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations. Math. Comput. 28 (1974), 145-162. DOI 10.2307/2005822 | MR 0331793 | Zbl 0309.65034
[30] K. Rektorys: The Method of Discretization in Time and Partial Differential Equations. Mathematics and Its Applications (East European Series) 4, D. Reidel Publishing, Dordrecht; SNTL-Publishers of Technical Literature, Praha (1982). MR 0689712 | Zbl 0505.65029
[31] F. Roskovec: Numerical solution of nonlinear convection-diffusion problems by adaptive methods. Master Thesis (2014), Czech.
[32] E. F. Toro: Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introduction, Springer, Berlin. (1999). DOI 10.1007/978-3-662-03915-1 | MR 1717819 | Zbl 0923.76004
[33] M. Vlasák, V. Dolejší, J. Hájek: A priori error estimates of an extrapolated space-time discontinuous Galerkin method for nonlinear convection-diffusion problems. Numer. Methods Partial Differ. Equations 27 (2011), 1456-1482. DOI 10.1002/num.20591 | MR 2838303 | Zbl 1237.65105
[34] M. Vlasák, Z. Vlasáková: Derivation of BDF coefficients for equidistant time step. Programs and Algorithms of Numerical Mathematics 15 Proc. Seminar, Dolní Maxov, Academy of Sciences of the Czech Republic, Institute of Mathematics, Praha (2010), 221-226. MR 3203769 | Zbl 1340.65137
[35] J. von Neumann: Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes. Math. Nachr. 4 (1951), 258-281. DOI 10.1002/mana.3210040124 | MR 0043386 | Zbl 0042.12301
[36] O. B. Widlund: A note on unconditionally stable linear multistep methods. BIT, Nord. Tidskr. Inf.-behandl. 7 (1967), 65-70. DOI 10.1007/BF01934126 | MR 0215533 | Zbl 0178.18502
[37] K. Wright: Some relationships between implicit Runge-Kutta, collocation Lanczos $\tau$ methods, and their stability properties. BIT, Nord. Tidskr. Inf.-behandl. 10 (1970), 217-227. DOI 10.1007/BF01936868 | MR 0266439 | Zbl 0208.41602

Affiliations:   Miloslav Vlasák, Charles University, Faculty of Mathematics and Physics, Department of Mathematical Analysis, Sokolovská 83, 186 75 Praha 8, Czech Republic, e-mail: vlasak@karlin.mff.cuni.cz

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