Mathematics of Computation

Avoiding the order reduction of Runge-Kutta methods for linear initial boundary value problems

Author(s): M. P. Calvo; C. Palencia.
Journal: Math. Comp. 71 (2002), 1529-1543.
MSC (2000): Primary 65M12, 65M20
Posted: November 19, 2001
Retrieve article in: PDF DVI PostScript
This article is available free of charge

Abstract: A new strategy to avoid the order reduction of Runge-Kutta methods when integrating linear, autonomous, nonhomogeneous initial boundary value problems is presented. The solution is decomposed into two parts. One of them can be computed directly in terms of the data and the other satisfies an initial value problem without any order reduction. A numerical illustration is given. This idea applies to practical problems, where spatial discretization is also required, leading to the full order both in space and time.

1.: S. Abarbanel, D. Gottlieb & M. H. Carpenter, On the removal of boundary errors caused by Runge-Kutta integration of nonlinear partial differential equations, SIAM J. Sci. Comput. 17 (1996) 777-782. MR 96m:65071
2.: I. Alonso-Mallo, Rational methods with optimal order of convergence for partial differential equations, Appl. Numer. Math. 35 (2000) 265-292. CMP 2001:05
3.: I. Alonso-Mallo, Explicit single step methods with optimal order of convergence for partial differential equations, Appl. Numer. Math. 31 (1999) 117-131. MR 2000d:65186
4.: I. Alonso-Mallo & C. Palencia, On the convolution operators arising in the study of abstract initial boundary value problems, Proc. Royal Soc. Edinburgh 126A (1996) 515-539. MR 97d:34068
5.: I. Alonso-Mallo & C. Palencia, Optimal orders of convergence in Runge-Kutta methods for linear, non-homogeneous PDEs with singular source terms, preprint.
6.: N. Yu. Bakaev, On the stability of some general discretization methods, Dokl. Akad. Nauk. SSSR 309 (1989) 11-15; English transl. in Soviet Math. Dokl. 40 (1990). MR 91e:65104
7.: N. Yu. Bakaev, Estimates for the resolvent of a multi-dimensional elliptic difference operator, Function.-Diff. Eqs., Perm' PPI (1991) 118-126. MR 94e:39001
8.: N. Yu. Bakaev, On variable stepsize Runge-Kutta approximations of a Cauchy problem for the evolution equation, BIT 38 (1998) 462-485. MR 99i:65069
9.: C. A. Brebbia, J. C. F. Telles & L. C. Wrobel, Boundary element techniques. Theory and applications in engineering, Springer, Berlin, 1984. MR 89i:65002
10.: P. Brenner, M. Crouzeix & V. Thomée, Single step methods for inhomogeneous linear differential equations in Banach space, R.A.I.R.O. Anal. Numer. 16 (1982) 5-26. MR 83d:65268
11.: P. Brenner & V. Thomée, On rational approximations of semigroups, SIAM J. Numer. Anal. 16 (1979) 683-694. MR 80j:47052
12.: M. H. Carpenter, D. Gottlieb, S. Abarbanel & W. S. Don, The theoretical accuracy of Runge-Kutta time discretizations for the initial boundary value problem: a study of the boundary error, SIAM J. Sci. Comput. 16 (1995) 1241-1252. MR 96h:65088
13.: M. Crouzeix & P. A. Raviart, Méthodes de Runge-Kutta, Unpublished Lecture Notes, Université de Rennes (1980).
14.: M. Crouzeix, S. Larsson, S. Piskarev & V. Thomée, The stability of rational approximations of holomorphic semigroups, BIT 33 (1993) 74-84. MR 96f:65069
15.: J. L. M. van Dorsselaer, J. F. B. M. Kraaijenvanger & M. N. Spijker, Linear stability analysis in the numerical solution of initial value problems, in Acta Numer. 1993, 199-237. MR 94e:65051
16.: E. Hairer & G. Wanner, Solving ordinary differential equations II, Stiff and differential-algebraic problems, 2nd Ed., Springer, Berlin, 1996. MR 97m:65007
17.: K. Ito & F. Kappel, The Trotter-Kato theorem and approximations of PDEs, Math. Comp. 67 (1998) 21-44. MR 98e:47060
18.: C. Johnson, Numerical solution of partial differential equations by the finite element method, Cambridge University Press, Cambridge, 1987. MR 89b:65003a
19.: S. L. Keeling, Galerkin/Runge-Kutta discretizations for parabolic equations with time-dependent coefficients, Math. Comp. 52 (1989) 561-586. MR 90a:65239
20.: S. L. Keeling, Galerkin/Runge-Kutta discretizations for semilinear parabolic equations, SIAM J. Numer. Anal. 27 (1990) 394-418. MR 91d:65140
21.: Ch. Lubich & A. Ostermann, Interior estimates for time discretizations of parabolic equations, Appl. Numer. Math. 18 (1995) 241-251. MR 96f:65124
22.: A. R. Mitchell & D. F. Griffiths, The finite difference method in partial differential equations, Wiley-Interscience, New York, 1980. MR 82a:65002
23.: K. W. Morton & D. F. Mayers, Numerical solution of partial differential equations, Cambridge University Press, Cambridge, 1994. MR 96b:65001
24.: A. Ostermann & M. Roche, Runge-Kutta methods for partial differential equations and fractional orders of convergence, Math. Comp. 59 (1992) 403-420. MR 93a:65125
25.: C. Palencia, A stability result for sectorial operators in Banach spaces, SIAM J. Numer. Anal. 30 (1993) 1373-1384. MR 94j:65109
26.: C. Palencia, On the stability of variable stepsize rational approximations of holomorphic semigroups, Math. Comput. 62 (1994) 93-103. MR 94c:47066
27.: C. Palencia, Maximum-norm analysis of completely discrete finite element methods for parabolic problems, SIAM J. Numer. Anal. 33 (1996) 1654-1668. MR 97e:65099
28.: C. Palencia & I. Alonso-Mallo, Abstract initial boundary value problems, Proc. Royal Soc. Edinburgh 124A (1994) 879-908. MR 95k:35090
29.: C. Palencia & J. M. Sanz-Serna, An extension of the Lax-Richtmyer theory, Numer. Math. 44 (1984) 279-283. MR 86c:65096
30.: D. Pathria, The correct formulation of intermediate boundary conditions for Runge-Kutta time integration of initial boundary value problems, SIAM J. Sci. Comput. 18 (1997) 1255-1266. MR 98d:65100
31.: A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer, Berlin, 1983. MR 85g:47061
32.: J. M. Sanz-Serna, J. G. Verwer & W. H. Hundsdorfer, Convergence and order reduction of Runge-Kutta schemes applied to evolutionary problems in partial differential equations, Numer. Math. 50 (1986) 405-418. MR 88f:65146
33.: A. H. Schatz & L. B. Wahlbin, On the quasi-optimality in $L_\infty$ of the -projection into finite element spaces, Math. Comp. 38 (1982) 1-22. MR 82m:65106
34.: A. H. Schatz, V. Thomée & L. B. Wahlbin, Stability, analyticity, and almost best approximation in maximum norm for parabolic finite element equations, Comm. Pure Appl. Math. 51 (1998) 1349-1385. MR 99h:65171
35.: B. Sz.-Nagy & C. Foias, Harmonic Analysis of Operators on Hilbert Spaces, North-Holland, Amsterdam, 1970. MR 43:947
36.: V. Thomée, Galerkin finite element methods for parabolic problems, Springer, Berlin, 1997. MR 98m:65007

M. P. Calvo
Affiliation: Departamento de Matemática Aplicada y Computación, Universidad de Valladolid, Valladolid, Spain
Email: maripaz@mac.cie.uva.es

C. Palencia
Affiliation: Departamento de Matemática Aplicada y Computación, Universidad de Valladolid, Valladolid, Spain
Email: palencia@mac.cie.uva.es

DOI: 10.1090/S0025-5718-01-01362-X
PII: S 0025-5718(01)01362-X
Keywords: Abstract initial boundary value problems, Runge-Kutta, order reduction
Received by editor(s): January 14, 2000
Received by editor(s) in revised form: November 30, 2000
Posted: November 19, 2001
Additional Notes: This research has been supported by DGICYT under project PB95-705 and by Junta de Castilla y León under project VA36/98.
Copyright of article: Copyright 2001, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6842(e) ISSN 0025-5718(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS