The present work constitutes the third installment in a series of investigations focused on discrete conformal structures on surfaces with boundary. In our preceding works [23, 24], we have established, respectively, a systematic classification of these discrete conformal structures and key results concerning their rigidity and existence. Building on this foundational work, the present study focuses on the deformation theory of discrete conformal structures on surfaces with boundary. Specifically, we introduce the combinatorial Ricci flow and the combinatorial Calabi flow for such structures, and establish the longtime existence and global convergence of solutions to these combinatorial curvature flows. These results provide effective algorithms for finding discrete hyperbolic metrics on surfaces with totally geodesic boundaries of prescribed lengths.
