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

• Published in: Journal of computational physics 2014-02, Vol.258, p.489-508
• Main Authors: Alldredge, Graham W., Hauck, Cory D., OʼLeary, Dianne P., Tits, André L.
• Format: Article
• Language: English
• Subjects: Algorithms; Boundaries; Closures; Computer simulation; Convex optimization; Entropy-based closures; Kinetic theory; Mathematical models; Moment equations; Optimization; Quadratures; Realizability; Regularization; Transport
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2013.10.049

Entropy-based (MN) 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.