Runge-Kutta time discretizations of nonlinear dissipative evolution equations

Author:
Eskil Hansen

Journal:
Math. Comp. **75** (2006), 631-640

MSC (2000):
Primary 65J15, 65M12

DOI:
https://doi.org/10.1090/S0025-5718-05-01866-1

Published electronically:
December 19, 2005

MathSciNet review:
2196983

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Global error bounds are derived for Runge-Kutta 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 infinite-dimensional spaces. An algebraically stable Runge-Kutta 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.**2.**M. G. Crandall,*Nonlinear semigroups and evolution governed by accretive operators*, Proc. Sympos. Pure Math.**45**(1986), pp. 305-337.**3.**K. Dekker and J. G. Verwer,*Stability of Runge-Kutta methods for stiff nonlinear differential equations*, North-Holland Publishing Co., Amsterdam, 1984.**4.**R. Frank, J. Schneid and C. W. Ueberhuber,*Order results for implicit Runge-Kutta methods applied to stiff systems*, SIAM J. Numer. Anal.**22**(1985), pp. 515-534.**5.**C. González, A. Ostermann, C. Palencia and M. Thalhammer,*Backward Euler discretization of fully nonlinear parabolic problems*, Math. Comp.**71**(2002), pp. 125-145.**6.**E. Hairer and G. Wanner,*Solving ordinary differential equations. II. Stiff and differential-algebraic problems*, Second edition, Springer-Verlag, 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. 823-841.**9.**Ch. Lubich and A. Ostermann,*Runge-Kutta methods for parabolic equations and convolution quadrature*, Math. Comp.**60**(1993), pp. 105-131.**10.**-,*Runge-Kutta approximation of quasilinear parabolic equations*, Math. Comp.**64**(1995), pp. 601-627.**11.**A. Ostermann and M. Thalhammer,*Convergence of Runge-Kutta methods for nonlinear parabolic equations*, Appl. Numer. Math.**42**(2002), pp. 367-380.**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. 389-407.**13.**A. Pazy,*Semigroups of linear operators and applications to partial differential equations*, Springer-Verlag, New York, 1983.**14.**G. Söderlind,*On nonlinear difference and differential equations*, BIT**24**(1984), pp. 667-680.**15.**-,*Bounds on nonlinear operators in finite-dimensional Banach spaces*, Numer. Math.**50**(1986), pp. 27-44.**16.**V. Thomée,*Galerkin finite element methods for parabolic problems*, Springer-Verlag, New York, 1997.**17.**E. Zeidler,*Nonlinear functional analysis and its applications. II/B. Nonlinear monotone operators*, Springer-Verlag, New York, 1990.

Retrieve articles in *Mathematics of Computation*
with MSC (2000):
65J15,
65M12

Retrieve articles in all journals with MSC (2000): 65J15, 65M12

Additional Information

**Eskil Hansen**

Affiliation:
Centre for Mathematical Sciences, Lund University, Box 118, SE-221 00 Lund, Sweden

Email:
eskil@maths.lth.se

DOI:
https://doi.org/10.1090/S0025-5718-05-01866-1

Keywords:
Nonlinear evolution equations,
logarithmic Lipschitz constants,
$m$-dissipative maps,
Runge-Kutta 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.