Convergence of implicit Runge-Kutta time discretisation methods for fundamental models in nonlinear acoustics

For the first time, a class of implicit Runge-Kutta time discretisation methods is studied for nonlinear damped wave equations arising in nonlinear acoustics. The analysis in particular applies to the Westervelt, Jordan-Moore-Gibson-Thompson, and Blackstock-Crighton-Brunnhuber-Jordan-Kuznetsov equations. Under appropriate regularity, consistency, and smallness requirements on the time-continuous solutions, global error bounds are obtained from energy estimates for the time-discrete solutions. Existence and uniqueness of time-discrete solutions as well as their convergence are proven under weaker conditions on the initial data, based on energy estimates that are established in a continuous setting and then transferred to the time discretisations.