Nonlinear geometric optics based multiscale stochastic Galerkin methods for highly oscillatory transport equations with random inputs

摘要

We develop generalized polynomial chaos (gPC) based stochastic Galerkin (SG)methods for a class of highly oscillatory transport equations that arise insemiclassical modeling of non-adiabatic quantum dynamics. These models containuncertainties, particularly in coefficients that correspond to the potentialsof the molecular system. We first focus on a highly oscillatory scalar modelwith random uncertainty. Our method is built upon the nonlinear geometricaloptics (NGO) based method, developed in cite{NGO} for numerical approximationsof deterministic equations, which can obtain accurate pointwise solution evenwithout numerically resolving spatially and temporally the oscillations. Withthe random uncertainty, we show that such a method has oscillatory higher orderderivatives in the random space, thus requires a frequency dependentdiscretization in the random space. We modify this method by introducing a new"time" variable based on the phase, which is shown to be non-oscillatory in therandom space, based on which we develop a gPC-SG method that can captureoscillations with the frequency-independent time step, mesh size as well as thedegree of polynomial chaos. A similar approach is then extended to asemiclassical surface hopping model system with a similar numerical conclusion.Various numerical examples attest that these methods indeed capture accuratelythe solution statistics {em pointwisely} even though none of the numericalparameters resolve the high frequencies of the solution.