An Asymptotic Parallel-in-Time Method for Highly Oscillatory PDEs

We present a new time-stepping algorithm for nonlinear PDEs that exhibit scale separation in time of a highly oscillatory nature. The algorithm combines the parareal method---a parallel-in-time scheme introduced in [J.-L. Lions, Y. Maday, and G. Turinici, C. R. Acad. Sci. Paris Ser. I Math., 332 (2001), pp. 661--668]---with techniques from the heterogeneous multiscale method (cf. [W. E and B. Engquist, Notices Amer. Math. Soc., 50 (2003), pp. 1062--1070]), which make use of the slow asymptotic structure of the equations [A. J. Majda and P. Embid, Theoret. Comput. Fluid Dyn., 11 (1998), pp. 155--169]. We present error bounds, based on the analysis in [M. J. Gander and E. Hairer, in Domain Decomposition Methods in Science and Engineering XVII, Springer, Berlin, 2008, pp. 45--56] and [G. Bal, in Domain Decomposition Methods in Science and Engineering, Springer, Berlin, 2005, pp. 425--432], that demonstrate convergence of the method. A complexity analysis also demonstrates that the parallel speedup increases ...

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