RungeKutta time discretizations of nonlinear dissipative evolution equations
Author:
Eskil Hansen
Journal:
Math. Comp. 75 (2006), 631640
MSC (2000):
Primary 65J15, 65M12
Published electronically:
December 19, 2005
MathSciNet review:
2196983
Fulltext PDF Free Access
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Abstract: Global error bounds are derived for RungeKutta time discretizations of fully nonlinear evolution equations governed by dissipative vector fields on Hilbert spaces. In contrast to earlier studies, the analysis presented here is not based on linearization procedures, but on the fully nonlinear framework of logarithmic Lipschitz constants in order to extend the classical convergence theory to infinitedimensional spaces. An algebraically stable RungeKutta method with stage order is derived to have a global error which is at least of order or , depending on the monotonicity properties of the method.
 1.
V. Barbu, Nonlinear semigroups and differential equations in Banach spaces, Noordhoff International Publishing, Leyden, 1976.
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M. G. Crandall, Nonlinear semigroups and evolution governed by accretive operators, Proc. Sympos. Pure Math. 45 (1986), pp. 305337.
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K. Dekker and J. G. Verwer, Stability of RungeKutta methods for stiff nonlinear differential equations, NorthHolland Publishing Co., Amsterdam, 1984.
 4.
R. Frank, J. Schneid and C. W. Ueberhuber, Order results for implicit RungeKutta methods applied to stiff systems, SIAM J. Numer. Anal. 22 (1985), pp. 515534.
 5.
C. González, A. Ostermann, C. Palencia and M. Thalhammer, Backward Euler discretization of fully nonlinear parabolic problems, Math. Comp. 71 (2002), pp. 125145.
 6.
E. Hairer and G. Wanner, Solving ordinary differential equations. II. Stiff and differentialalgebraic problems, Second edition, SpringerVerlag, Berlin, 1996.
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E. Hansen, Convergence of multistep time discretizations of nonlinear dissipative evolution equations, To appear in SIAM J. Numer. Anal.
 8.
I. Higueras and G. Söderlind, Logarithmic norms and nonlinear DAE stability, BIT 42 (2002), pp. 823841.
 9.
Ch. Lubich and A. Ostermann, RungeKutta methods for parabolic equations and convolution quadrature, Math. Comp. 60 (1993), pp. 105131.
 10.
, RungeKutta approximation of quasilinear parabolic equations, Math. Comp. 64 (1995), pp. 601627.
 11.
A. Ostermann and M. Thalhammer, Convergence of RungeKutta methods for nonlinear parabolic equations, Appl. Numer. Math. 42 (2002), pp. 367380.
 12.
A. Ostermann, M. Thalhammer and G. Kirlinger, Stability of linear multistep methods and applications to nonlinear parabolic problems, Appl. Numer. Math. 48 (2004), pp. 389407.
 13.
A. Pazy, Semigroups of linear operators and applications to partial differential equations, SpringerVerlag, New York, 1983.
 14.
G. Söderlind, On nonlinear difference and differential equations, BIT 24 (1984), pp. 667680.
 15.
, Bounds on nonlinear operators in finitedimensional Banach spaces, Numer. Math. 50 (1986), pp. 2744.
 16.
V. Thomée, Galerkin finite element methods for parabolic problems, SpringerVerlag, New York, 1997.
 17.
E. Zeidler, Nonlinear functional analysis and its applications. II/B. Nonlinear monotone operators, SpringerVerlag, New York, 1990.
 1.
 V. Barbu, Nonlinear semigroups and differential equations in Banach spaces, Noordhoff International Publishing, Leyden, 1976.
 2.
 M. G. Crandall, Nonlinear semigroups and evolution governed by accretive operators, Proc. Sympos. Pure Math. 45 (1986), pp. 305337.
 3.
 K. Dekker and J. G. Verwer, Stability of RungeKutta methods for stiff nonlinear differential equations, NorthHolland Publishing Co., Amsterdam, 1984.
 4.
 R. Frank, J. Schneid and C. W. Ueberhuber, Order results for implicit RungeKutta methods applied to stiff systems, SIAM J. Numer. Anal. 22 (1985), pp. 515534.
 5.
 C. González, A. Ostermann, C. Palencia and M. Thalhammer, Backward Euler discretization of fully nonlinear parabolic problems, Math. Comp. 71 (2002), pp. 125145.
 6.
 E. Hairer and G. Wanner, Solving ordinary differential equations. II. Stiff and differentialalgebraic problems, Second edition, SpringerVerlag, Berlin, 1996.
 7.
 E. Hansen, Convergence of multistep time discretizations of nonlinear dissipative evolution equations, To appear in SIAM J. Numer. Anal.
 8.
 I. Higueras and G. Söderlind, Logarithmic norms and nonlinear DAE stability, BIT 42 (2002), pp. 823841.
 9.
 Ch. Lubich and A. Ostermann, RungeKutta methods for parabolic equations and convolution quadrature, Math. Comp. 60 (1993), pp. 105131.
 10.
 , RungeKutta approximation of quasilinear parabolic equations, Math. Comp. 64 (1995), pp. 601627.
 11.
 A. Ostermann and M. Thalhammer, Convergence of RungeKutta methods for nonlinear parabolic equations, Appl. Numer. Math. 42 (2002), pp. 367380.
 12.
 A. Ostermann, M. Thalhammer and G. Kirlinger, Stability of linear multistep methods and applications to nonlinear parabolic problems, Appl. Numer. Math. 48 (2004), pp. 389407.
 13.
 A. Pazy, Semigroups of linear operators and applications to partial differential equations, SpringerVerlag, New York, 1983.
 14.
 G. Söderlind, On nonlinear difference and differential equations, BIT 24 (1984), pp. 667680.
 15.
 , Bounds on nonlinear operators in finitedimensional Banach spaces, Numer. Math. 50 (1986), pp. 2744.
 16.
 V. Thomée, Galerkin finite element methods for parabolic problems, SpringerVerlag, New York, 1997.
 17.
 E. Zeidler, Nonlinear functional analysis and its applications. II/B. Nonlinear monotone operators, SpringerVerlag, New York, 1990.
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Additional Information
Eskil Hansen
Affiliation:
Centre for Mathematical Sciences, Lund University, Box 118, SE221 00 Lund, Sweden
Email:
eskil@maths.lth.se
DOI:
http://dx.doi.org/10.1090/S0025571805018661
PII:
S 00255718(05)018661
Keywords:
Nonlinear evolution equations,
logarithmic Lipschitz constants,
$m$dissipative maps,
RungeKutta methods,
algebraic stability,
$B$convergence
Received by editor(s):
December 14, 2004
Published electronically:
December 19, 2005
Article copyright:
© Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
