Error analysis of implicit Euler methods for quasilinear hyperbolic evolution equations

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

“…Much of our analysis (and also that of [8,10,12,17]) relies on state dependent norms. Let v, w ∈ B Y (R).…”

“…These solutions then automatically fulfill the divergence conditions (12c) and (12d). To tackle the problem in our framework, we set In [8] and [17] wave and Maxwell equations on a domain with Dirichlet boundary conditions were studied in a similar way. The Maxwell system with the usual boundary conditions of a perfect conductor can be treated in our framework only for special nonlinearities, see Proposition 4.8 in [17].…”

“…In the case of Maxwell's equations one cannot treat perfectly conducting boundaries without very strong restrictions on the nonlinearities, [17]. On the other hand, one can handle Dirichlet boundary conditions for quasilinear Maxwell and wave equations, see [17] and [8]. This shortcoming is unfortunate since the more operator-theoretic approach in [11,12,17] fits very well to the tools from numerical analysis used in this paper and also in [8,14].…”

“…The framework of [17] was already used in the recent paper [8] by two of the present authors, who have proved well-posedness and first order convergence of the semi-implicit and implicit Euler approximations to quasilinear hyperbolic evolution equations. These results have been applied to certain quasilinear Maxwell and wave equations.…”

“…On the other hand, one can handle Dirichlet boundary conditions for quasilinear Maxwell and wave equations, see [17] and [8]. This shortcoming is unfortunate since the more operator-theoretic approach in [11,12,17] fits very well to the tools from numerical analysis used in this paper and also in [8,14]. In future work we want to combine the present approach with more PDE type methods as in [1,5], for instance, to cover also Maxwell equations on domains with standard boundary conditions.…”