Runge–Kutta methods and viscous wave equations

We study the numerical time integration of a class of viscous wave equations by means of Runge–Kutta methods. The viscous wave equation is an extension of the standard second-order wave equation including advection–diffusion terms differentiated in time. The viscous wave equation can be very stiff so that for time integration traditional explicit methods are no longer efficient. A-Stable Runge–Kutta methods are then very good candidates for time integration, in particular diagonally implicit ones. Special attention is paid to the question how the A-Stability property can be translated to this non-standard class of viscous wave equations.