Adaptive change of basis in entropy-based moment closures for linear kinetic equations

TitleAdaptive change of basis in entropy-based moment closures for linear kinetic equations
Publication TypeJournal Article
Year of Publication2014
AuthorsAlldredge, GW, Hauck, CD, O'Leary, DP, Tits, AL
JournalJournal of Computational Physics
Pages489 - 508
Date Published2014/02/01

Entropy-based (M_N) moment closures for kinetic equations are defined by a
constrained optimization problem that must be solved at every point in a
space-time mesh, making it important to solve these optimization problems
accurately and efficiently. We present a complete and practical numerical
algorithm for solving the dual problem in one-dimensional, slab geometries. The
closure is only well-defined on the set of moments that are realizable from a
positive underlying distribution, and as the boundary of the realizable set is
approached, the dual problem becomes increasingly difficult to solve due to
ill-conditioning of the Hessian matrix. To improve the condition number of the
Hessian, we advocate the use of a change of polynomial basis, defined using a
Cholesky factorization of the Hessian, that permits solution of problems nearer
to the boundary of the realizable set. We also advocate a fixed quadrature
scheme, rather than adaptive quadrature, since the latter introduces
unnecessary expense and changes the computationally realizable set as the
quadrature changes. For very ill-conditioned problems, we use regularization to
make the optimization algorithm robust. We design a manufactured solution and
demonstrate that the adaptive-basis optimization algorithm reduces the need for
regularization. This is important since we also show that regularization slows,
and even stalls, convergence of the numerical simulation when refining the
space-time mesh. We also simulate two well-known benchmark problems. There we
find that our adaptive-basis, fixed-quadrature algorithm uses less
regularization than alternatives, although differences in the resulting
numerical simulations are more sensitive to the regularization strategy than to
the choice of basis.

Short TitleJournal of Computational Physics