Adaptive change of basis in entropybased moment closures for linear kinetic equations
Abstract
Entropybased (M_{N}) moment closures for kinetic equations are defined by a constrained optimization problem that must be solved at every point in a spacetime 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 onedimensional, slab geometries. The closure is only welldefined 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 illconditioning 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 illconditioned problems, we use regularization to make the optimization algorithm robust. We design a manufactured solution and demonstrate that the adaptivebasis 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 spacetime mesh. We also simulate two wellknown benchmark problems. There we find that our adaptivebasis, fixedquadrature 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.
 Publication:

Journal of Computational Physics
 Pub Date:
 February 2014
 DOI:
 10.1016/j.jcp.2013.10.049
 arXiv:
 arXiv:1306.2881
 Bibcode:
 2014JCoPh.258..489A
 Keywords:

 Physics  Computational Physics;
 Mathematics  Numerical Analysis
 EPrint:
 doi:10.1016/j.jcp.2013.10.049