The Boltzmann Transport Equation (BTE) is a linear integro-differential equation to
be solved for a scalar unknown, usually called the particle flux. Finite volume
schemes, on the other hand, are typically used to solve equations of high speed
fluid dynamics where conservation is crucial. Application of these schemes to
the BTE ensures not only conservation, but also provides for a more natural
handling of boundary conditions, material properties and source terms, as well
as an easier parallel implementation and post processing. The finite volume
scheme also lends itself to an efficient implementation of high order spatial
discretizations.
Material interfaces and time-dependent large source terms can introduce
severe oscillations even with second order fixed stencil schemes. Slope limiting,
or {\it essentially non-oscillatory} (ENO) spatial interpolations eliminate these
oscillations, and make higher-than-second-order spatial accuracies possible. A
newer variation of these nonlinear schemes is the Weighted ENO (WENO)
scheme that makes the stencil transition less abrupt and boosts the accuracy in
smooth regions. 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. Both of these
approaches require a preconditioner in order to obtain a reasonable rate of
convergence.
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.
This work was performed under the auspices of the U.S. Department of Energy by
University of California, Lawrence Livermore National Laboratory under Contract
W-7405-Eng-48.
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