A layer‐adaptive numerical scheme for time‐fractional weakly singular interface problem modeling dendritic solidification processes

Abstract The interface problem arising from the dendritic solidification of pure substances is governed by the distribution of heat in a phase‐changing medium. This study develops an efficient layer‐adapted difference scheme for a mathematical interface model formulated as a time‐fractional convection‐diffusion problem featuring a discontinuous convection coefficient. The fractional derivative of order is interpreted in the Caputo sense. The computational domain is divided into two subdomains, each governed by a time‐fractional convection‐diffusion equation. Due to the mild singularity at , the solution typically exhibits an initial layer, which adversely affects the accuracy of standard polynomial interpolation on uniform meshes. To overcome this challenge and ensure optimal convergence, we employ the L1 discretization for the fractional derivative on a nonuniform graded mesh. The spatial derivatives are approximated using a second‐order finite difference scheme. The convergence analysis is carried out using the discrete comparison principle along with a suitably chosen barrier function, leading to rigorous error estimates. Numerical experiments are provided to support the theoretical results, confirming the accuracy, efficiency, and robustness of the proposed scheme.