Error analysis of implicit Runge–Kutta methods for quasilinear hyperbolic evolution equations

Abstract

We establish error bounds of implicit Runge-Kutta methods for a class of quasilinear hyperbolic evolution equations including certain Maxwell and wave equations. Our assumptions cover algebraically stable and coercive schemes such as Gauß and Radau collocation methods. We work in a refinement of the analytical setting of Kato's well-posedness theory.

Finite element error analysis of wave equations with dynamic boundary conditions:L2 estimates

“…We first show stability for implicit Runge-Kutta methods applied to the ODE system (7.5). The proof of the stability bound is a straightforward simplification of the corresponding results of (Mansour, 2015, Lemma 4.1), or (Hochbruck & Pažur, 2015, Section 3), (Hochbruck et al, 2018, Lemma 5.1-5.2), (Kovács & Lubich, 2018, where more general problems than (7.5) are considered.…”

“…Some parts have already been covered in the literature, only the L 2 norm requires some simple modifications. We give these details, but for those parts which are not new we only give detailed references, following Mansour (2015); Hochbruck & Pažur (2015); Hochbruck et al (2018); Kovács & Lubich (2018). We strongly believe, and the previous references also strengthen, that these techniques extend to time discretisations of more general, e.g.…”

Diagonally implicit Runge-Kutta schemes: Discrete energy-balance laws and compactness properties

“…As a final comment we note that the mathematical theory about Galerkin-in-time and/or full-tableau RK methods developed by the numerical PDE community; see [41,42,1,61,35,22] and references therein, rarely ever applies (or even mentions) DIRK schemes. We do however, highlight the preexistence of quite relevant material with specific focus on DIRK schemes that shares a few common points of intersection with the material presented in this manuscript.…”

“…Its major drawback, however, is that it is only first order accurate. In order to remedy this issue, higher order generalizations, like Galerkin-in-time schemes, have been developed and analyzed; see for instance [41,42,1,61,35,60,22] and references therein. These schemes can also be shown to possess energy-balance laws, and are of arbitrary high order.…”

Exponential Integrators for Quasilinear Wave-Type Equations

“…However, one can actually also find approximation rates of the implicit and semi-implicit Euler method in there. By completely new techniques, these results could be improved to optimal order in [15] for equations of the type (1.1) and higher order Runge-Kutta methods were discussed in [14,21]. In the case of the onedimensional wave equation equipped with periodic boundary condition, error bounds for semi-discretization in time and full discretization were proved in [9].…”