The phase flow method

This paper introduces the phase flow method, a novel, accurate and fast approach for constructing phase maps for nonlinear autonomous ordinary differential equations. The method operates by initially constructing the phase map for small times using a standard ODE integration rule and builds up the phase map for larger times with the help of a local interpolation scheme together with the group property of the phase flow. The computational complexity of building up the complete phase map is usually that of tracing a few rays. In addition, the phase flow method is provably and empirically very accurate. Once the phase map is available, integrating the ODE for initial conditions on the invariant manifold only makes use of local interpolation, thus having constant complexity. The paper develops applications in the field of high frequency wave propagation, and shows how to use the phase flow method to 1) rapidly propagate wave fronts, 2) rapidly calculate wave amplitudes along these wave fronts, and 3) rapidly evaluate multiple wave arrival times at arbitrary locations.