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Travis Austin and Thomas Manteuffel, "A Least-Squares Method for the Linear Boltzmann Equation with Anisotropic Scattering: Theoretical Results", LA-UR-04-2325 submitted to SINUM, 2004


Least-squares methods have been applied to a wide range of differential equations and have been established to be competitive with other existing discretization strategies [Bochev and Gunzburger, SIAM Review, Vol.~40]. In this article, we consider a least-squares method for the linear Boltzmann equation with anisotropic scattering. A similar method has already been developed, and extensively examined, for the linear Boltzmann equation with isotropic scattering. The success of the least-squares method for isotropic scattering depends on scaling the linear Boltzmann equation so that minimization of the least-squares functional in a discrete space always yields accurate discrete solutions. A similar scaling of the linear Boltzmann equation is employed for anisotropic scattering. In the previous work for isotropic scattering, coercivity and continuity results were established for the scaled least-squares functional relative to a physically reasonable norm. In this paper, we extend the previous coercivity and continuity results so that they hold in this more general case of anisotropic scattering under reasonable assumptions on the coefficients of the scattering kernel. Additionally, we extend the bounds for the discretization error for the thin regime and for the thick regime. For the thick regime, we establish optimal error estimates for the case of mildly anisotropic scattering and highly anisotropic scattering.

BibTeX Entry

author = {Travis Austin and Thomas Manteuffel},
title = {A Least-Squares Method for the Linear Boltzmann Equation with Anisotropic Scattering: Theoretical Results},
year = {2004},
urlpdf = {http://math.lanl.gov/~austint/Papers/austin-2004-aniso-ls.pdf},
note = { LA-UR-04-2325 submitted to SINUM}