Search

We introduce a new numerical strategy to solve a class of oscillatory transport partial differential equation (PDE) models which is able to capture accurately the solutions without numerically resolving the high frequency oscillations in both space and time. Such PDE models arise in semiclassical modeling of quantum dynamics with band-crossings, and other highly oscillatory waves. Our first main idea is to use the geometric optics ansatz, which builds the oscillatory phase into an independent variable. We then choose suitable initial data, based on the Chapman-Enskog expansion, for the new model. For a scalar model, we prove that so constructed models will have certain smoothness, and consequently, for a first-order approximation scheme we prove uniform error estimates independent of the (possibly small) wavelength. The method is extended to systems arising from a semiclassical model for surface hopping, a non-adiabatic quantum dynamic phenomenon. Numerous numerical examples demonstrate that the method has the desired properties. [ABSTRACT FROM AUTHOR]

In this paper, we build a two-scale macro-micro decomposition of the Vlasov equation with a strong magnetic field. This consists in writing the solution of this equation as a sum of two oscillating functions with circumscribed oscillations. The first of these functions has a shape which is close to the shape of the two-scale limit of the solution and the second one is a correction built to offset this imposed shape. The aim of such a decomposition is to be the starting point for the construction of two-scale asymptotic-preserving schemes. [ABSTRACT FROM AUTHOR]