Volume 54, Number 6, November-December 2020
|Page(s)||1849 - 1882|
|Published online||16 September 2020|
Nonlinear geometric optics based multiscale stochastic Galerkin methods for highly oscillatory transport equations with random inputs
Univ Rennes, Inria Rennes & Institut de Recherche Mathématiques de Rennes, CNRS UMR 6625 Rennes & ENS Rennes, Rennes, France
2 School of Mathematical Sciences, Institute of Natural Sciences, MOE-LSC and SHL-MAC, Shanghai Jiao Tong University, Shanghai, P.R. China
3 Univ Rennes, CNRS & Institut de Recherche Mathématiques de Rennes, CNRS UMR 6625 & Inria Rennes & ENS Rennes, Rennes, France
4 Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong SAR
Accepted: 19 December 2019
We develop generalized polynomial chaos (gPC) based stochastic Galerkin (SG) methods for a class of highly oscillatory transport equations that arise in semiclassical modeling of non-adiabatic quantum dynamics. These models contain uncertainties, particularly in coefficients that correspond to the potentials of the molecular system. We first focus on a highly oscillatory scalar model with random uncertainty. Our method is built upon the nonlinear geometrical optics (NGO) based method, developed in Crouseilles et al. [Math. Models Methods Appl. Sci. 23 (2017) 2031–2070] for numerical approximations of deterministic equations, which can obtain accurate pointwise solution even without numerically resolving spatially and temporally the oscillations. With the random uncertainty, we show that such a method has oscillatory higher order derivatives in the random space, thus requires a frequency dependent discretization 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 the random space, based on which we develop a gPC-SG method that can capture oscillations with the frequency-independent time step, mesh size as well as the degree of polynomial chaos. A similar approach is then extended to a semiclassical surface hopping model system with a similar numerical conclusion. Various numerical examples attest that these methods indeed capture accurately the solution statistics pointwisely even though none of the numerical parameters resolve the high frequencies of the solution.
Mathematics Subject Classification: 35Q40 / 35L03 / 4Q10 / 65M12
Key words: Highly oscillatory PDEs / nonlinear geometric optics / asymptotic preserving / uncertainty quantification / generalized polynomial chaos / stochastic Galerkin method / surface hopping
© EDP Sciences, SMAI 2020
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