Which transmission condition is the best in optimized Schwarz waveform relaxation for viscoelastic equation: Robin or characteristic?

The viscoelastic equation introduces a viscoelastic damping term in the classical wave equation, thus exhibits properties of parabolic systems accompanied by hyperbolic properties. On the other hand, it is well known that for parabolic systems, when solved using the optimized Schwarz waveform relaxation (OSWR) method, Robin transmission condition is preferred, however characteristic transmission condition is good for hyperbolic system. In this paper, we investigate what transmission condition should be applied according to different strengths of damping. The results show that, in the OSWR algorithm, for weak damping the characteristic transmission condition is preferred, while for strong damping one should use Robin instead, as anticipated. It is also observed that the convergence depends on the relationship between the temporal and spatial mesh sizes applied, which shows overlap may not accelerate the convergence. In addition, we find numerically that a mix of both transmission conditions can only improve the convergence for strong damping, and it remains to be an interesting problem to look for transmission conditions that are more robust in the damping strength.