Melnik, R.V.N. and Melnik, K.N.
In this paper we analyze propblems of mathematical modeling in microelectronics on the basis of a hierarchy constrated with respect to the relaxation time functions. For the analysis of non-equilibrium and non-local processes in semiconductors, effective numerical procedures based on non-conservative monotone schemeshave been developed, algorithmic realization of the proposed schemes have been proposed, and results of computational experiments on device modeling have been presented.
From a mathematical point of view, the models considered in this paper provide an important practical example of stiff systems of partial differential equations (PDEs) with source terms. The availabilityof the conservative property for such systems depends on the definition of the source terms which are always a subject of approximations. If perturbation techniques are applied to such problems, the natural space for perturbations become L1 rather than L2. A challenging problem arises from the fact that the flow map representing the solution of such systems is typically not differentiable with respect to the linaer structure of L1. This leads to majoe difficulties in the analysis of the systems that are described by stiff PDEs, because the contractivity property of the flow for such systems with respect to L1-distance is no longer true in general. As a result, reasonable alternatives to classical and continuous mathematical models for semiconductor device modeling become discrete physics-based topological models, semi-classical, and quantum models. In the development of such models, the role of effective computational techniques essentially increases.
Key words: kinetic, hydrodynamic, and quasi-hydrodynamic models; non-equilibrium dynamics; semiconductor modelling; accounting for non-local effects in mathematical models; hierarchy of mathematical models; relaxation time; maximum principle; exponential difference schemes; conservation laws; non-conservative models; non-reflexive Banach spaces; L1-distance; monotonicity requirements; numerical analysis; discrete physics-based topological models; semi-classical and quantum models.
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