Mathematics Colloquia and Seminars

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The Boltzmann Transport Equation (BTE) is a linear integro-differential equation to be solved for the scalar unknown $\Psi$, usually called the {\it particle flux}. Material interfaces and time-dependent, spatially discontinous large source terms can introduce severe oscillations even with second order fixed stencil schemes. Slope limiting, or {\it essentially nonoscillatory} (ENO) spatial interpolations eliminate these oscillations, and make higher-than-second-order spatial accuracies possible. For unsteady problems, the resulting nonlinear spatial discretization yields a set of ODE's in time, which in turn is solved via high order implicit time-stepping with error control. For the steady-state case, we need to solve the non-linear system, typically by Newton-Krylov iterations.We will discuss the advantages of using an ENO/WENO method, as well as the various issues introduced by such nonlinear methods originally designed for computing shocked fluid flows. There will be several numerical examples presented to demonstrate the accuracy, non-oscillatory nature and efficiency of these high order methods, in comparison with other fixed-stencil schemes. Parallel efficiency, scalability, boundary conditions and convergence acceleration aspects will be addressed as well, concluding with work in progress and open problems.