Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Runge-Kutta methods for parabolic equations and convolution quadrature

Authors: Ch. Lubich and A. Ostermann
Journal: Math. Comp. 60 (1993), 105-131
MSC: Primary 65M12; Secondary 65D30, 65M15, 65R20, 76D05, 76M25
MathSciNet review: 1153166
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the approximation properties of Runge-Kutta time discretizations of linear and semilinear parabolic equations, including incompressible Navier-Stokes equations. We derive asymptotically sharp error bounds and relate the temporal order of convergence, which is generally noninteger, to spatial regularity and the type of boundary conditions. The analysis relies on an interpretation of Runge-Kutta methods as convolution quadratures. In a different context, these can be used as efficient computational methods for the approximation of convolution integrals and integral equations. They use the Laplace transform of the convolution kernel via a discrete operational calculus.

References [Enhancements On Off] (What's this?)

  • [1] P. Brenner, M. Crouzeix, and V. Thomée, Single step methods for inhomogeneous linear differential equations in Banach space, RAIRO Modél. Math. Anal. Numér. 16 (1982), 5-26. MR 648742 (83d:65268)
  • [2] K. Burrage, W. H. Hundsdorfer, and J. G. Verwer, A study of B-convergence of Runge-Kutta methods, Computing 36 (1986), 17-34. MR 832927 (87i:65102)
  • [3] M. Crouzeix, Sur l'approximation des équations différentielles opérationelles linéaires par des méthodes de Runge-Kutta, Thèse d'Etat, Université Paris VI, 1975.
  • [4] M. Crouzeix and P. A. Raviart, Approximation des problèmes d'évolution, unpublished lecture notes, Université de Rennes, 1980.
  • [5] P. P. B. Eggermont and Ch. Lubich, Uniform error estimates of operational quadrature methods for nonlinear convolution equations on the half-line, Math. Comp. 56 (1991), 149-176. MR 1052091 (91j:65195)
  • [6] H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal. 16 (1964), 269-315. MR 0166499 (29:3774)
  • [7] D. Fujiwara, Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order, Proc. Japan Acad. 43 (1967), 82-86. MR 0216336 (35:7170)
  • [8] P. Grisvard, Caractérisation de quelques espaces d'interpolation, Arch. Rational Mech. Anal. 25 (1967), 40-63. MR 0213864 (35:4718)
  • [9] E. Hairer and G. Wanner, Solving ordinary differential equations II. Stiff and differential-algebraic problems, Springer-Verlag, Berlin and Heidelberg, 1991. MR 1111480 (92a:65016)
  • [10] D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Math., vol. 840, Springer-Verlag, 1981. MR 610244 (83j:35084)
  • [11] J. G. Heywood and R. Rannacher, Finite-element approximation of the nonstationary Navier-Stokes problem. Part IV: Error analysis for second-order time discretization, SIAM J. Numer. Anal. 27 (1990), 353-384. MR 1043610 (92c:65133)
  • [12] O. A. Karakashian, On Runge-Kutta methods for parabolic problems with time-dependent coefficients, Math. Comp. 47 (1986), 77-101. MR 842124 (87i:65161)
  • [13] S. L. Keeling, Galerkin/Runge-Kutta discretizations for semilinear parabolic equations, SIAM J. Numer. Anal. 27 (1990), 394-418. MR 1043612 (91d:65140)
  • [14] I. Lasiecka, Galerkin approximations of abstract parabolic boundary value problems with rough boundary data--$ {L_p}$ theory, Math. Comp. 47 (1986), 55-75. MR 842123 (87i:65187)
  • [15] M. N. Le Roux, Semidiscretization in time for parabolic problems, Math. Comp. 33 (1979), 919-931. MR 528047 (80f:65101)
  • [16] Ch. Lubich, Convolution quadrature and discretized operational calculus. I, II, Numer. Math. 52 (1988), 129-145, 413-425. MR 923707 (89g:65018)
  • [17] -, On the convergence of multistep methods for nonlinear stiff differential equations, Numer. Math. 58 (1991), 839-853; Erratum, 61 (1992), 277-279. MR 1098868 (92d:65127)
  • [18] Ch. Lubich and O. Nevanlinna, On resolvent conditions and stability estimates, BIT 31 (1991), 293-313. MR 1112225 (92h:65145)
  • [19] A. Ostermann and M. Roche, Runge-Kutta methods for partial differential equations and fractional orders of convergence, Math. Comp. 59 (1992), 403-420. MR 1142285 (93a:65125)
  • [20] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983. MR 710486 (85g:47061)
  • [21] B. A. Schmitt, Stability of implicit Runge-Kutta methods for nonlinear stiff differential equations, BIT 28 (1988), 884-897. MR 972813 (90a:65187)
  • [22] V. Thomée, Galerkin finite element methods for parabolic problems, Lecture Notes in Math., vol. 1054, Springer-Verlag, 1984.
  • [23] J. G. Verwer, Convergence and order reduction of diagonally implicit Runge-Kutta schemes in the method of lines, Numerical Analysis (D. F. Griffiths and G. A. Watson, eds.), Pitman Res. Notes Math. Ser., vol. 140, Longman Sci. Tech., Harlow, 1986, pp. 220-237. MR 873112 (88f:65158)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65M12, 65D30, 65M15, 65R20, 76D05, 76M25

Retrieve articles in all journals with MSC: 65M12, 65D30, 65M15, 65R20, 76D05, 76M25

Additional Information

Keywords: Parabolic equations, nonstationary Navier-Stokes equation, Runge-Kutta time discretization, convolution integrals, numerical quadrature
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society