The High Order Augmented Finite Volume Methods Based on Series Expansion for Nonlinear Degenerate Parabolic Equations.

Two high order multi-augmented and improved augmented finite volume methods are proposed for solving nonlinear degenerate parabolic problems. The solution is represented as Puiseux series expansion in a subdomain with singularity, but contains undetermined parameters called augmented variables. The equation in regular subdomain is treated with high accuracy numerical methods and the unknown parameters can be solved simultaneously from the corresponding nonlinear system. The outstanding advantages of the proposed methods are that the degeneracy can be depicted by the semi-analytic solution, and we can get high order results globally. Specially, the convergence order for nonlinear degenerate parabolic problems is determined by the numerical schemes on regular subdomain, and the augmented methods have good robustness for solving degenerate or singular problems. Numerical examples for some degenerate parabolic equations confirm the efficiency of the new methods including the second and fourth order schemes. In particular, a two-dimensional singular elliptic equation with corner degeneracy is presented in the numerical experiments to demonstrate that the proposed methods can be extended to higher dimensional degenerate problems. [ABSTRACT FROM AUTHOR]