Gauss-Runge-Kutta time discretization of wave equations on evolving surfaces (Q2514239)

An error analysis for a full discretization of the linear wave equation on a moving surface is presented. The moving surface is a compact moving hypersurface \(\Gamma(t) \subset \mathbb{R}^m\), \(t \in [0,T]\), with a given velocity \(v(x,t)\). The paper is an extension of results obtained in a previous work of several authors, namely by using a Gauss-Runge-Kutta time discretization. For space discretization the evolving surface finite element method developed by \textit{G. Dziuk} and \textit{C. M. Elliott} [IMA J. Numer. Anal. 27, No. 2, 262--292 (2007; Zbl 1120.65102)] is used. Previous works used numerical solutions as piecewise linear functions in space and time, and they are stable under a CFL condition, which brings a proper choice for the time step. To avoid this restriction and to obtain higher-order accuracy in time, fully implicit Gauss-Runge-Kutta methods are used. Optimal error estimates are established in the natural time-dependent norms under sufficient regularity of the exact solution \(u\) of the wave equation. To achieve this goal, mainly algebraic stability, the coercivity property of the Gauss-Runge-Kutta methods and the properties of space discretization by the evolving surface finite element method are used. Finally, numerical experiments that support theoretical results concerning error bounds and unconditional stability conclude the paper.