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

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.