A High-precision Augmented Variable Numerical Method For Nonlinear Degenerate Parabolic Equations

This thesis proposes multi-augmented variables and improved augmented variable numerical methods based on Puiseux series for solving nonlinear degenerate parabolic equations.Each method gives two numerical schemes which are second order in time,second order and fourth order in space,respectively.The solution is represented as Puiseux series expansion in the singular domain with singularity,but contains unknown augmented variables.The equation in the regular domain is treated with finite volume and compact finite volume methods.The unknown parameters in the Puiseux series and solutions in the regular domain can be solved simultaneously from the coupled nonlinear equation.The advantages of the proposed methods are that the singularity can be treated with Puiseux series,so we can get high order numerical results globally.Specially,the improved augmented variable numerical method is easy to implement,with the universality of the method being extended and the computation efficiency being improved.As to the degenerate parabolic problem,the convergence orders of the two methods are determined by the numerical schemes on the regular domain,so we give two numerical schemes which are second order in both time and space and second order in time,fourth order in space,respectively.Numerical examples confirm the efficiency of the proposed two methods,and we give an example of two-dimensional degenerate elliptic equation to make a preliminary attempt to extend the methods to two-dimensional degenerate problems.