Stability and convergence of time discretizations of quasi-linear evolution equations of Kato type
Provides rigorous numerical analysis for a broad class of PDEs, but the results are theoretical and incremental (extending known techniques to a specific framework).
The paper proves stability and optimal-order convergence of implicit time discretizations (midpoint rules and higher-order Runge-Kutta methods) for quasi-linear evolution equations of Kato type, which include symmetric hyperbolic systems and fluid/wave equations.
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.