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Title: Runge–Kutta time discretization of nonlinear parabolic equations studied via discrete maximal parabolic regularity
Authors: Kunstmann, PC
Li, B 
Lubich, C
Issue Date: Oct-2018
Source: Foundations of computational mathematics, Oct. 2018, p. 1109-1130
Abstract: For a large class of fully nonlinear parabolic equations, which include gradient flows for energy functionals that depend on the solution gradient, the semidiscretization in time by implicit Runge–Kutta methods such as the Radau IIA methods of arbitrary order is studied. Error bounds are obtained in the (Formula presented.) norm uniformly on bounded time intervals and, with an improved approximation order, in the parabolic energy norm. The proofs rely on discrete maximal parabolic regularity. This is used to obtain (Formula presented.) estimates, which are the key to the numerical analysis of these problems.
Keywords: Error bounds
Gradient flow
Maximal parabolic regularity
Nonlinear parabolic equation
Runge–Kutta method
Stability
Publisher: Springer
Journal: Foundations of computational mathematics 
ISSN: 1615-3375
DOI: 10.1007/s10208-017-9364-x
Rights: © SFoCM 2017
This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Nature’s AM terms of use (https://www.springernature.com/gp/open-research/policies/accepted-manuscript-terms), but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: http://dx.doi.org/10.1007/s10208-017-9364-x
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