Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

Runge-Kutta methods for partial differential equations and fractional orders of convergence


Authors: A. Ostermann and M. Roche
Journal: Math. Comp. 59 (1992), 403-420
MSC: Primary 65M20; Secondary 65L06, 65M12
DOI: https://doi.org/10.1090/S0025-5718-1992-1142285-6
MathSciNet review: 1142285
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We apply Runge-Kutta methods to linear partial differential equations of the form $ {u_t}(x,t) = \mathcal{L}(x,\partial )u(x,t) + f(x,t)$. Under appropriate assumptions on the eigenvalues of the operator $ \mathcal{L}$ and the (generalized) Fourier coefficients of f, we give a sharp lower bound for the order of convergence of these methods. We further show that this order is, in general, fractional and that it depends on the $ {L^r}$-norm used to estimate the global error. The analysis also applies to systems arising from spatial discretization of partial differential equations by finite differences or finite element techniques. Numerical examples illustrate the results.


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 Anal. Numér. 16 (1982), 5-26. MR 648742 (83d:65268)
  • [2] P. Brenner, V. Thomée, and L. B. Wahlbin, Besov spaces and applications to difference methods for initial value problems. Lecture Notes in Math., vol. 434, Springer-Verlag, Berlin-Heidelberg, 1975. MR 0461121 (57:1106)
  • [3] K. Burrage and W. H. Hundsdorfer, The order of B-convergence of algebraically stable Runge-Kutta methods, BIT 27 (1987), 62-71. MR 874861 (88e:65081)
  • [4] 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)
  • [5] J. C. Butcher, The numerical analysis of ordinary differential equations: Runge-Kutta and general-linear methods, Wiley, Chichester, 1987. MR 878564 (88d:65002)
  • [6] M. Crouzeix, Sur l'approximation des équations différentielles opérationelles linéaires par des méthodes de Runge-Kutta, Thèse d'Etat, Université Paris VI, 1975.
  • [7] M. Crouzeix and P. A. Raviart, Méthodes de Runge-Kutta, Unpublished Lecture Notes, Université de Rennes, 1980.
  • [8] R. Frank, J. Schneid, and C. W. Ueberhuber, The concept of B-convergence, SIAM J. Numer. Anal. 18 (1981), 753-780. MR 629662 (82h:65054)
  • [9] E. Hairer, Ch. Lubich, and M. Roche, Error of Runge-Kutta methods for stiff problems studied via differential algebraic equations, BIT 28 (1988), 678-700. MR 963310 (90e:65101)
  • [10] E. Hairer, S. P. Nørsett, and G. Wanner, Solving ordinary differential equations I. Nonstiff problems, Springer-Verlag, Berlin-Heidelberg, 1987. MR 868663 (87m:65005)
  • [11] E. Hairer and G. Wanner, Solving ordinary differential equations II, Stiff and differential-algebraic problems, Springer-Verlag, Berlin-Heidelberg, 1991. MR 1111480 (92a:65016)
  • [12] M. N. Le Roux, Semidiscretization in time for parabolic problems, Math. Comp. 33 (1979), 919-931. MR 528047 (80f:65101)
  • [13] -, Méthodes multipas pour des équations paraboliques non linéaires, Numer. Math. 35 (1980), 143-162. MR 585243 (81i:65075)
  • [14] Ch. Lubich, On the convergence of multistep methods for nonlinear stiff differential equations, Numer. Math. 58 (1991), 839-853, and Corrigendum, ibid. 61 (1992), 277-279. MR 1098868 (92d:65127)
  • [15] A. Ostermann and M. Roche, Rosenbrock methods for partial differential equations and fractional orders of convergence, Submitted for publication.
  • [16] A. Prothero and A. Robinson, On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations, Math. Comp. 28 (1974), 145-162. MR 0331793 (48:10125)
  • [17] J. M. Sanz-Serna and J. G. Verwer, Stability and convergence at the PDE/Stiff ODE interface, Appl. Numer. Math. 5 (1989), 117-132. MR 979551 (90c:65126)
  • [18] J. M. Sanz-Serna, J. G. Verwer, and W. H. Hundsdorfer, Convergence and order reduction of Runge-Kutta schemes applied to evolutionary problems in partial differential equations, Numer. Math. 50 (1987), 405-418. MR 875165 (88f:65146)
  • [19] S. Scholz, Order barriers for the B-convergence of SDIRK methods, Preprint, TU Dresden, 1987.
  • [20] H. Triebel, Interpolation theory, function spaces, differential operators, North-Holland, Amsterdam, 1978. MR 503903 (80i:46032b)
  • [21] 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 Research Notes in Math., vol. 140, 1986, pp. 220-237. MR 873112 (88f:65158)
  • [22] R. M. Young, An introduction to nonharmonic Fourier series, Academic Press, New York, 1980. MR 591684 (81m:42027)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65M20, 65L06, 65M12

Retrieve articles in all journals with MSC: 65M20, 65L06, 65M12


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1992-1142285-6
Keywords: Runge-Kutta methods, method of lines, partial differential equations
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society