The Efficient Implicit Asymptotic Preserving Scheme In The Diffusive Regime Of Linear Boltzmann Equation

It is important to simulate the Boltzmann equation from both the aspects of un-derstanding the mesoscopic physical mechanism and the practical applications. Atthe mesoscopic level, the Botlzmann type kinetic equation can accurately describe themesoscopic physical phenomena, while the macroscopic equations such as hydrody-namic or difusion equation cannot. Examples include neutron transport, thermal ra-diation, the difusion of the electron in the semiconductor and so on. However，theKnudsen number in the Boltzmann equation often difers in several orders within onephysical domain which causes the numerical stifness and the inefciency of the stan-dard numerical methods. The asymptotic-preserving (AP) scheme is one of the multi-scale simulation methods that has attracted lots of attention recently. The method hasthe uniform stability and accuracy with respect to the Knudsen number in the equation.When the Knudsen number goes to zero, the AP scheme is able to capture the correctasymptotic behavior even with coarse grids and large time steps, and thus avoids thedetermination of the coupling interface, including both the location and the conditionas in other multiscale and multiphysics computational methods.The thesis takes the linear semiconductor Boltzmann equation (LSBE) as themodel problem to develop the AP scheme through the BGK-penalty method, firstintroduced by Filbet and Jin, and the operator-splitting technique. We derive fourAP schemes, which are the time-splitting difusive relaxation scheme (TSDRS), thetime-unsplitting difusive relaxation scheme (TUDRS), the time-splitting implicit APscheme based on the uniform grids (IMUG), and the time-splitting implicit AP schemebased on the staggered grids (IMSG), respectively. All of the schemes can capture thecorrect difusive limits of the LSBE with large mesh sizes and time steps. Accordingto the Von-Neumann analysis, TSDRS and TUDRS applied to the Goldstein-Taylormodel are shown to be uniformly stable with respect to the parabolic conditions, andmeanwhile, IMUG and IMSG applied to the Goldstein-Taylor model are uncondition- ally stable. These are the first class of asymptotic-preserving schemes ever introducedthat have the following property: the time step does not need to be the square of themesh size (the parabolic stability condition) even in the difusive regime, yet one doesnot need to invert the non-local collision operators. Large number of numerical exper-iments show that the schemes are able to capture the correct physical solutions evenwith the coarse mesh grids and large time steps, and thus improve the computationefciency significantly and greatly reduce the computational complexity.