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.
Mots-clés : Highly oscillatory PDEs, nonlinear geometric optics, asymptotic preserving, uncertainty quantification, generalized polynomial chaos, stochastic Galerkin method, surface hopping
@article{M2AN_2020__54_6_1849_0,
author = {Crouseilles, Nicolas and Jin, Shi and Lemou, Mohammed and Liu, Liu},
title = {Nonlinear geometric optics based multiscale stochastic {Galerkin} methods for highly oscillatory transport equations with random inputs},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1849--1882},
publisher = {EDP-Sciences},
volume = {54},
number = {6},
year = {2020},
doi = {10.1051/m2an/2019094},
mrnumber = {4150228},
zbl = {1471.65137},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2019094/}
}
TY - JOUR AU - Crouseilles, Nicolas AU - Jin, Shi AU - Lemou, Mohammed AU - Liu, Liu TI - Nonlinear geometric optics based multiscale stochastic Galerkin methods for highly oscillatory transport equations with random inputs JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2020 SP - 1849 EP - 1882 VL - 54 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2019094/ DO - 10.1051/m2an/2019094 LA - en ID - M2AN_2020__54_6_1849_0 ER -
%0 Journal Article %A Crouseilles, Nicolas %A Jin, Shi %A Lemou, Mohammed %A Liu, Liu %T Nonlinear geometric optics based multiscale stochastic Galerkin methods for highly oscillatory transport equations with random inputs %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2020 %P 1849-1882 %V 54 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2019094/ %R 10.1051/m2an/2019094 %G en %F M2AN_2020__54_6_1849_0
Crouseilles, Nicolas; Jin, Shi; Lemou, Mohammed; Liu, Liu. Nonlinear geometric optics based multiscale stochastic Galerkin methods for highly oscillatory transport equations with random inputs. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 6, pp. 1849-1882. doi : 10.1051/m2an/2019094. http://www.numdam.org/articles/10.1051/m2an/2019094/
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