SUMMARYInadequate spatial mesh resolution for simulation of the time-dependent Ginzburg-Landau equations is shown to give rise to spurious solutions. Phenomenological studies to examine the effect of the physical parameters and boundary conditions in 2D and 3D are presented to illustrate the solution structure and to highlight non-linear effects related to evolving vortex patterns. We illustrate and explore this issue further by considering a related simplified model in which the number of vortices is equal to the 'winding number' that is associated with the applied boundary conditions. Using this model we demonstrate that the solution structure is non-unique for several values of winding number.
In this article we concisely present several modern strategies that are applicable to driftdominated
carrier transport in higher-order deterministic models such as the driftdiffusion,
hydrodynamic, and quantum hydrodynamic systems. The approaches include
extensions of “upwind” and artificial dissipation schemes, generalization of the traditional
Scharfetter – Gummel approach, Petrov – Galerkin and streamline-upwind Petrov
Galerkin (SUPG), “entropy” variables, transformations, least-squares mixed methods
and other stabilized Galerkin schemes such as Galerkin least squares and discontinuous
Galerkin schemes. The treatment is representative rather than an exhaustive
review and several schemes are mentioned only briefly with appropriate reference
to the literature. Some of the methods have been applied to the semiconductor device
problem while others are still in the early stages of development for this class of applications.
We have included numerical examples from our recent research tests with
some of the methods. A second aspect of the work deals with algorithms that employ
unstructured grids in conjunction with adaptive refinement strategies. The full benefits
of such approaches have not yet been developed in this application area and we
emphasize the need for further work on analysis, data structures and software to
support adaptivity. Finally, we briefly consider some aspects of software frameworks.
These include dial-an-operator approaches such as that used in the industrial simulator
PROPHET, and object-oriented software support such as those in the SANDIA
National Laboratory framework SIERRA.
We examine multigrid solution for convection–diffusion problems in which convective effects are significant. In particular, we consider the case where the fine grid satisfies the associated cell or diagonal dominance condition but coarse grid levels violate this condition. Related numerical experiments are conducted. We also introduce an alternative local elliptic projection technique and compare this with interpolating the error corrections from coarse to fine grids. In addition, we demonstrate the use of grid redistribution and optimization techniques in the multilevel context.
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