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Title: Optimal error estimates for high order Runge-Kutta methods applied to evolutionary equations

Abstract

Fully discrete approximations to 1-periodic solutions of the Generalized Korteweg de-Vries and the Cahn-Hilliard equations are analyzed. These approximations are generated by an Implicit Runge-Kutta method for the temporal discretization and a Galerkin Finite Element method for the spatial discretization. Furthermore, these approximations may be of arbitrarily high order. In particular, it is shown that the well-known order reduction phenomenon afflicting Implicit Runge Kutta methods does not occur. Numerical results supporting these optimal error estimates for the Korteweg-de Vries equation and indicating the existence of a slow motion manifold for the Cahn-Hilliard equation are also provided.

Authors:
Publication Date:
Research Org.:
Tennessee Univ., Knoxville, TN (United States)
OSTI Identifier:
5241289
Resource Type:
Miscellaneous
Resource Relation:
Other Information: Thesis (Ph. D.)
Country of Publication:
United States
Language:
English
Subject:
99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; KORTEWEG-DE VRIES EQUATION; RUNGE-KUTTA METHOD; ERRORS; GALERKIN-PETROV METHOD; MATHEMATICAL MANIFOLDS; PROBABILISTIC ESTIMATION; DIFFERENTIAL EQUATIONS; EQUATIONS; ITERATIVE METHODS; NUMERICAL SOLUTION; PARTIAL DIFFERENTIAL EQUATIONS; 990200* - Mathematics & Computers

Citation Formats

McKinney, W R. Optimal error estimates for high order Runge-Kutta methods applied to evolutionary equations. United States: N. p., 1989. Web.
McKinney, W R. Optimal error estimates for high order Runge-Kutta methods applied to evolutionary equations. United States.
McKinney, W R. 1989. "Optimal error estimates for high order Runge-Kutta methods applied to evolutionary equations". United States.
@article{osti_5241289,
title = {Optimal error estimates for high order Runge-Kutta methods applied to evolutionary equations},
author = {McKinney, W R},
abstractNote = {Fully discrete approximations to 1-periodic solutions of the Generalized Korteweg de-Vries and the Cahn-Hilliard equations are analyzed. These approximations are generated by an Implicit Runge-Kutta method for the temporal discretization and a Galerkin Finite Element method for the spatial discretization. Furthermore, these approximations may be of arbitrarily high order. In particular, it is shown that the well-known order reduction phenomenon afflicting Implicit Runge Kutta methods does not occur. Numerical results supporting these optimal error estimates for the Korteweg-de Vries equation and indicating the existence of a slow motion manifold for the Cahn-Hilliard equation are also provided.},
doi = {},
url = {https://www.osti.gov/biblio/5241289}, journal = {},
number = ,
volume = ,
place = {United States},
year = {Sun Jan 01 00:00:00 EST 1989},
month = {Sun Jan 01 00:00:00 EST 1989}
}

Miscellaneous:
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