Space-frequency adaptive approximation for quantum hydrodynamic models
by Silvia Bertoluzza and Paola Pietra
in Trans. Theory and Stat. Phys. 29 (2000), 375-395.
ABSTRACT
Numerical simulations of semiconductor Quantum Hydrodynamic models with uniform discretizations require an extremely high number of grid points, also when the ``pathology'' of the solution, which enforces the mesh size, is localized in a small percentage of the simulation domain. This leads to unnecessarily time consuming computations.
For an efficient discretization of QHD one has to take into account the two major characteristics of the solution of the system.
Due to the steep gradient of the doping profile and to the shape of the external potential, the electron position density or some of its derivatives may present strong variation or even blow up in some points.
Even more important, the dispersive character of the system implies that the solution may present high frequency oscillations, which are localized in regions not a priori known.
Therefore, an efficient approximation demands the use of a discretization where not only the spatial grid, but also the frequency distribution is adaptively adjusted to the behaviour of the solution.
One way of achieving such a goal is to use bases with good localization both in space and frequency. In particular wavelet type bases, which display such a property, have already been successfully applied in the design of efficient adaptive schemes in various application fields. Due to their characteristcs, the definition of criteria for driving the adaptive procedure (refining and coarsening) both in space and in frequency is quite natural.
Here we present a comparative study of several of such bases (wavelets, wavelet packets, local-sine bases, ...), aiming at identifying the ones better suited for an efficient approximation of the QHD solution.
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