Influence of the phase accuracy of the coarse solver calculation on the convergence of the parareal method iteration for hyperbolic PDEs

Gander and Petcu (ESAIM Proc 25:114–129, 2008) reported that, theoretically, the convergence of the parareal method iteration for hyperbolic PDEs is strongly influenced by the phase (frequency) accuracy of the coarse solver calculation. However, no numerical study has clearly shown this. Therefore, through numerical tests, we investigate the influence of the phase accuracy of the coarse solver calculation on the convergence of the parareal method iteration for hyperbolic PDEs. First, we consider a simple harmonic motion and a multi-DOF mass-spring system (MDMSS) as examples of hyperbolic PDEs using the modified Newmark-\(\beta \) method (Mizuta et al. in J JSCE 268:15–21, 1977), which can provide the exact phase of the time integration of a simple harmonic motion. Based on the results of the numerical tests, we show that the convergence of the parareal method iteration for hyperbolic PDEs is approximately independent of the parameters of parallel-in-time integration (PinT) and instead is dependent primarily on the phase accuracy of the coarse solver calculation. In addition, we show that reducing the number of bases in the reduced basis method (RBM) (Chen et al., in: Rozza (ed) Reduced order methods for modeling and computational reduction, MS and a modeling, simulation and applications, vol 9, Springer, Berlin, pp 187–214, 2014) causes the saturation of a decrease in an error during the parareal iteration for the MDMSS using the mode analysis method. The RBM is expected to make available accurate phase calculation in the coarse solver by maintaining the time step width as same as that of the fine solver. Second, we investigate whether the same saturation appears for the linear advection–diffusion equation when we use the RBM. We use the time evolution basis method in the RBM for the linear advection–diffusion equation. As a result, we show that reducing the number of bases causes the saturation of the decrease in the error in the linear advection–diffusion equation. Based on the results of the present study, an increase in the phase accuracy of the coarse solver calculation is strongly required for better convergence of the parareal method iteration for hyperbolic PDEs. Moreover, the saturation of the decrease in the error during the parareal method iteration should be overcome when using the RBM.