A parareal approach of semi-linear parabolic equations based on general waveform relaxation

We present a parareal approach of semi-linear parabolic equations based on general waveform relaxation (WR) at the partial differential equation (PDE) level. An algorithm for initial-boundary value problem and two algorithms for time-periodic boundary value problem are constructed. The convergence analysis of three algorithms are provided. The results show that the algorithm for initial-boundary value problem is superlinearly convergent while both algorithms for the time-periodic boundary value problem linearly converge to the exact solutions at most. Numerical experiments show that the parareal algorithms based on general WR at the PDE level, compared with the parareal algorithm based on the classical WR at the ordinary differential equations (ODEs) level (the PDEs is discretized into ODEs), require much fewer number of iterations to converge.