Lagrangian Numerical Approximations to One-Dimensional Convolution-Diffusion Equations

This work focuses on the numerical analysis of one-dimensional nonlinear diffusion equations involving a convolution product. First, homogeneous friction equations are considered. Algorithms follow recent ideas on mass transportation methods and lead to simple schemes which can be proved to be stable, to decrease entropy, and to converge toward the unique solution of the continuous problem. In particular, for the first time, homogeneous cooling states are displayed numerically. Further, we present results on the more delicate fourth-order thin-film equation for which a nonnegativity-preserving scheme is derived. The dead core phenomenon is presented for the Hele–Shaw cell.