Error analysis of implicit Runge-Kutta methods for quasilinear hyperbolic evolution equations (Q2413466)

The authors study implicit Runge-Kutta methods for the quasilinear hyperbolic evolution equation \(\Lambda (u(t)) u_t (t)=Au(t)+Q(u(t)) u(t)\), \(u(0)=u_0\) in a Hilbert space setting. Here, \(A\) is a linear skew-adjoint operator, \(\Lambda(v)\) is a symmetric positive definite operator for \(v\) in a neighborhood of zero and \(Q\) denotes a lower order term. The setting includes certain Maxwell and wave equations on entire space or under Dirichlet boundary conditions. Error bounds for a class of Runge-Kutta methods are established. The results cover algebraically stable and coercive schemes such as Gauss and Radau collocation methods.