Numer. Math. (2018) 138:557–579
Error analysis of implicit Runge–Kutta methods for
quasilinear hyperbolic evolution equations
· Tomislav Pažur
· Roland Schnaubelt
Received: 6 March 2017 / Revised: 4 August 2017 / Published online: 28 August 2017
© Springer-Verlag GmbH Deutschland 2017
Abstract We establish error bounds of implicit Runge–Kutta methods for a class
of quasilinear hyperbolic evolution equations including certain Maxwell and wave
equations on full space or with Dirichlet boundary conditions. Our assumptions cover
algebraically stable and coercive schemes such as Gauß and Radau collocation meth-
ods. We work in a reﬁnement of the analytical setting of Kato’s well-posedness theory.
Mathematics Subject Classiﬁcation Primary: 65M12, 65J15; Secondary: 35Q61,
Quasilinear hyperbolic evolution equations describe a wide range of phenomena in
physics, including in particular the Maxwell system with nonlinear constitutive laws.
There is a well established analytical theory for such problems. On the other hand,
despite their importance, for quasilinear hyperbolic problems there are only very few
rigorous convergence results concerning time integration methods. The implicit Euler
This work was supported by the Deutsche Forschungsgemeinschaft (DFG) via CRC 1173.
Department of Mathematics, Karlsruhe Institute of Technology, 76149 Karlsruhe, Germany
AVL AST d.o.o. Croatia, 10020 Zagreb, Croatia