Lagrangian numerical approximations to one-dimensional convolution-diffusion equations

Laurent Gosse (1) , Giuseppe Toscani (2)

1 IAC - Istituto per le Applicazioni del Calcolo "Mauro Picone"
2 UNIPV - Dipartimento di Matematica “Felice Casorati” Department of Mathematics [Univ Pavia]

This work focuses on the numerical analysis of 1D 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-films equation for which a nonnegativity-preserving scheme is derived. Dead core phenomenon is presented for the Hele--Shaw cell.

Mots clés: Hele--Shaw cell; Granular flows; friction equations; Lagrangian approximation; Wasserstein metric; homogeneous cooling states; Hele--Shaw cell.

Domaines: Analyse numérique [math.NA]; Mécanique des fluides [physics.class-ph]; Mécanique des fluides [physics.class-ph]

Laurent Gosse, Giuseppe Toscani. Lagrangian numerical approximations to one-dimensional convolution-diffusion equations. SIAM Journal on Scientific Computing, 2006, 28 (4), pp.1203 - 1227. ⟨10.1137/050628015⟩. ⟨hal-00426881⟩