Numerical Analysis of Augmented FVM for Nonlinear Time Fractional Degenerate Parabolic Equation

Utilizing the nonlinear Time Fractional Degenerate Parabolic Equation (TFDPE) in modeling provides a comprehensive approach to studying phenomena exhibiting both fractional order dynamics and degenerate parabolic behavior, facilitating accurate predictions and insights across diverse scientific domains. However, the numerical solution of TFDPE is a challenging task and traditional numerical methods cannot solve this equation because of the spatial singularity influence. In this paper, we find the numerical solution of nonlinear TFDPE with both strongly and weakly degenerate cases using the higher order augmented finite volume method on uniform grids. To handle the singularity of TFDPE, we choose an intermediate point near the singular point and split the whole domain into singular and regular subdomains. Then, we find the solution on singular subdomain using the Puiseux series while on the regular subdomain we find the solution by finite volume schemes. The main idea is to recover the Puiseux series on singular subdomain using the Picard iteration methods which is also a challenging because of the time fractional derivative in the original equation. The solution on the singular subdomain is in the form of Puiseux series, which has multiple undetermined augmented variables and these variables play a role in organically combining the singular and regular subdomains. To approximate the time fractional derivative, we use the second order weighted and shifted Grünwald difference (WSGD) scheme and give the comparison of our results with $L_1$-scheme. We use the discrete energy method to prove that the schemes has temporal second order while spatial second and fourth-order on the whole domain and for the augmented variables in discrete $L_2$-norm. Finally, we give some numerical examples to confirm the accuracy and order of convergence of the proposed schemes for the whole domain and the augmented variables. We also give an interesting example with coefficient blow-up at the degenerate point and show the schemes are working the same as the other cases.