This work presents a method to quantitatively calculate the error induced through application of approximate boundary conditions in quantum charge transport simulations based on the Wigner transport equation (WTE). Except for the special case of homogeneous material, there exists no methodology for the calculation of exact boundary conditions. Consequently, boundary conditions are customarily approximated by equilibrium or near-equilibrium distributions known to be correct in the classical limit. This practice can, however, exert deleterious impact on the accuracy of numerical calculations and can even lead to unphysical results. The Yoder group has recently developed a series expansion for exact boundary conditions which, when truncated, can be used to calculate boundary conditions of successively greater accuracy through consideration of successively higher order terms, the computational penalty for which is however not to be underestimated. This thesis focuses on the calculation and analysis of the second order term of the series expansion. A method is demonstrated to calculate the term for any general device structure in one spatial dimension. In addition, numerical analysis is undertaken to directly compare the first and second order terms. Finally a method to incorporate the first order term into simulation is formulated.

The Swedish defence agency (FOI) has developed a particle model (called Pello) that simulates the dispersion of aerosols and gases. At the boundaries, such as the ground, the particles can either reflect back into the domain (the atmosphere) or be absorbed. Which of the events that occurs is decided by a certain probability, which in the present model depends on mere physical properties. In this thesis we have investigated a newly proposed boundary behaviour which also depends on the time step used in the numerical simulations. We verified the accuracy of the new model by using a dispersion model with an explicit solution. To gain a better understanding of how important parameters at the boundary influence each other, we performed a sensitivity analysis. Simulations showed an overall improving concentration profile as the time step became smaller and the new model working well. The convergence order of the simulations was found to be close to 0.5. In this thesis we have shown that there exist an upper limit for the time step, which depends on the specific model. The present used time step at FOI does not have this versatile property. But having this upper limit for the time step close to the boundary, and a uniform time step can be time demanding. This lead us to the conclusion that an adaptive time step should be implemented.

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We study the optimal regularity and nondegeneracy of a free boundary problem related to the fractional Laplacian through the extension technique of Caffarelli and Silvestre. Specifically, we show that minimizers of the energy [mathematical equation] where [mathematical equations] with 0 < [gamma] < 1, with free behavior on the set {y=0}, are Holder continuous with exponent [Beta] = 2[sigma]/2-[gamma]. These minimizers exhibit a free boundary: along {y = 0}, they divide into a zero set {u = 0} and a positivity set where {u > 0}; we call the interface between these sets the free boundary. The regularity is optimal, due to the non-degeneracy property of the minimizers: in any ball of radius r centered at the free boundary, the minimizer grows (in the supremum sense) like r[Beta]. This work is related to, but addresses a different problem from, recent work of Caffarelli, Roquejoffre, and Sire. / text

We analyze the geometry and regularity of the largest subsolution of a Free Boundary Problem. We showed that the largest subsolution is a viscosity solution of (1) with Lipschitz and Non-Degenerate properties under a very general free boundary condition. In addition to this, we provide density bounds for the positivity set and its complement near the free boundary. / text

published_or_final_version / abstract / toc / Mechanical Engineering / Doctoral / Doctor of Philosophy

Under consideration are two-point boundary value problems for a system of second order differential equations which contains a small parameter multiplying the highest dereivatives. We prove the existence of solutions exhibiting left and right boundary layers by constructing upper and lower solutions of the system. The behavior of the solutions as the parameter tends to zero is also established. Of special interest is the existence of a compound boundary layer (i.e., one involving two scales) at the left endpoint of the interval.

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No description available.