PARAREAL CONVERGENCE FOR OSCILLATORY PDEs WITH FINITE TIME-SCALE SEPARATION

In [SIAM J. Sci. Comput., 36 (2014), pp. A693-A713] the authors present a new coarse propagator for the parareal method applied to oscillatory PDEs that exhibit time-scale separation and show, under certain regularity constraints, superlinear convergence which leads to significant parallel speedups over standard parareal methods. The error bound depends on the degree of time-scale separation, epsilon, and the coarse time step, Delta T, and relies on a bound that holds only in the limit of small epsilon. The main result of the present paper is a generalization of this error bound that also holds for finite values of epsilon, which can be important for applications in the absence of scale separation. The new error bound is found to depend on an additional parameter, eta, the averaging window used in the nonlinear term of the coarse propagator. The new proof gives insight into how the parareal method can converge even for finite values of epsilon. It is also a significant technical advance over the proof presented in [SIAM J. Sci. Comput., 36 (2014), pp. A693-A713]; it requires the introduction of a stiffness regulator function that allows us to control the oscillatory stiffness in the nonlinear term. The new convergence concepts developed in the new proof are confirmed using numerical simulations.