A-stable Runge-Kutta methods for semilinear evolution equations

TL;DR

This paper establishes the existence and smoothness of solutions for semilinear evolution equations and demonstrates that certain implicit Runge-Kutta methods are also smooth and convergent under specific conditions, including for wave and Schrödinger equations.

Contribution

It proves the smoothness of solutions and Runge-Kutta semidiscretizations for semilinear evolution equations within a Hilbert space framework, including convergence results.

Findings

Solutions are temporally smooth in the lowest scale norm.

Implicit A-stable Runge-Kutta methods are smooth maps between scales.

Full order convergence is achieved for normal or sectorial linear parts.

Abstract

We consider semilinear evolution equations for which the linear part generates a strongly continuous semigroup and the nonlinear part is sufficiently smooth on a scale of Hilbert spaces. In this setting, we prove the existence of solutions which are temporally smooth in the norm of the lowest rung of the scale for an open set of initial data on the highest rung of the scale. Under the same assumptions, we prove that a class of implicit, $A$-stable Runge--Kutta semidiscretizations in time of such equations are smooth as maps from open subsets of the highest rung into the lowest rung of the scale. Under the additional assumption that the linear part of the evolution equation is normal or sectorial, we prove full order convergence of the semidiscretization in time for initial data on open sets. Our results apply, in particular, to the semilinear wave equation and to the nonlinear Schr\"odinger equation.