Runge-Kutta time discretization of nonlinear parabolic equations studied via discrete maximal parabolic regularity

TL;DR

This paper analyzes the use of implicit Runge-Kutta methods, including Radau IIA, for time discretization of nonlinear parabolic equations, providing uniform error bounds and leveraging discrete maximal parabolic regularity for rigorous analysis.

Contribution

It introduces a novel error analysis framework for fully nonlinear parabolic equations discretized by Runge-Kutta methods, utilizing discrete maximal parabolic regularity.

Findings

Error bounds in $W^{1,inity}$ norm are established.

Improved approximation order in the parabolic energy norm.

Analysis applies to a broad class of nonlinear parabolic equations.

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 $W^{1,\infty}$ 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 $W^{1,\infty}$ estimates, which are the key to the numerical analysis of these problems.