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.

提出了两种求解高阶非线性退化抛物线问题的高阶多增幅和改进的有限体积方法。该解决方案表示为具有奇异性的子域中的Puiseux级数展开，但包含未确定的参数，称为增强变量。用高精度数值方法处理规则子域中的方程，并且可以从相应的非线性系统中同时求解未知参数。所提出方法的突出优点是可以用半解析解来描述简并性，并且可以在全球范围内获得高阶结果。特别地，非线性简并抛物线问题的收敛阶由规则子域上的数值方案确定，增强方法对退化或奇异问题具有很好的鲁棒性。一些退化的抛物线方程的数值示例证实了包括二阶和四阶格式的新方法的效率。特别地，在数值实验中提出了具有角退化的二维奇异椭圆方程，证明了所提出的方法可以推广到高维退化问题。