Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

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


Authors: M. P. Calvo and C. Palencia
Journal: Math. Comp. 71 (2002), 1529-1543
MSC (2000): Primary 65M12, 65M20
DOI: https://doi.org/10.1090/S0025-5718-01-01362-X
Published electronically: November 19, 2001
MathSciNet review: 1933043
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


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

  • 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 $H_1$-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

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65M12, 65M20

Retrieve articles in all journals with MSC (2000): 65M12, 65M20


Additional Information

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: https://doi.org/10.1090/S0025-5718-01-01362-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
Published electronically: 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.
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society