Two numerical methods are proposed and analyzed for one-dimensional nonlinear weakly degenerate parabolic equation with variable degenerate factors and a degenerate point at \(x=0\) . The presence of variable degenerate factors in the space direction poses a significant challenge, as it brings the logarithmic singularity for the solution. The novelty of the proposed method is to obtain the Puiseux series expansion of the solution which has an algebraic-logarithmic singularity at the degenerate point. With the integral expression of the solution, we utilize an iteration method to recover the Puiseux series expansion. This expansion contains an undetermined function, named as an augmented variable. Notably, the Puiseux series recovery technique can also be effectively applied to the blow-up and strongly degenerate problems. Since the Puiseux series can characterize the degeneracy within the singular domain, the weakly degenerate problem can be solved by coupling a uniform grid on a regular region with standard numerical schemes. Consequently, we propose an augmented finite volume method (AFVM) and an augmented compact finite volume method (ACFVM). A major advantage of the proposed schemes is that the convergence order of the globally ill-posed problem is determined by the convergence order of the well-posed problem on the regular subdomain. A rigorous error analysis for ACFVM is conducted and fourth-order accuracy in the spatial direction is obtained. The second-order and fourth-order accuracy of the proposed AFVM and ACFVM are validated through two numerical examples.
