Stability and convergence of time discretizations of quasi-linear evolution equations of Kato type

TL;DR

This paper analyzes the stability and convergence of various time discretization methods for quasi-linear evolution equations of Kato type, demonstrating optimal results under regularity conditions.

Contribution

It establishes stability and convergence results for implicit and Runge-Kutta methods applied to Kato-type equations, extending understanding of numerical solutions for hyperbolic systems.

Findings

Linearly implicit and fully implicit midpoint rules are stable and convergent.

Higher-order algebraically stable Runge-Kutta methods achieve optimal convergence.

Results apply to symmetric hyperbolic systems and fluid/wave equations.

Abstract

Semidiscretization in time is studied for a class of quasi-linear evolution equations in a framework due to Kato, which applies to symmetric first-order hyperbolic systems and to a variety of fluid and wave equations. In the regime where the solution is suffciently regular, we show stability and optimal-order convergence of the linearly implicit and fully implicit midpoint rules and of higher-order implicit Runge{Kutta methods that are algebraically stable and coercive, such as the collocation methods at Gauss nodes.