We show how existing models for the sedimentation of monodisperse flocculated suspensions and of polydisperse suspensions of rigid spheres differing in size can be combined to yield a new theory of the sedimentation processes of polydisperse suspensions forming compressible sediments ("sedimentation with compression"' or "sedimentation-consolidation process"). For N solid particle species, this theory reduces in one space dimension to an $N\times N$ coupled system of quasi-linear degenerate convection-diffusion equations. Analyses of the characteristic polynomials of the Jacobian of the convective flux vector and of the diffusion matrix show that this system is of strongly degenerate parabolic-hyperbolic type for arbitrary N and particle size distributions. Bounds for the eigenvalues of both matrices are derived. The mathematical model for N=3$ is illustrated by a numerical simulation obtained by the Kurganov-Tadmor central difference scheme for convection-diffusion problems. The numerical scheme exploits the derived bounds on the eigenvalues to keep the numerical diffusion to a minimum.

We consider scalar conservation laws with the spatially varying flux $H(x)f(u)+(1-H(x))g(u)$, where $H(x)$ is the Heaviside function and f and g are smooth nonlinear functions. Adimurthi, Mishra, and Veerappa Gowda [J. Hyperbolic Differ. Equ., 2 (2005), pp. 783–837] pointed out that such a conservation law admits many $L^1$ contraction semigroups, one for each so-called connection $(A,B)$. Here we define entropy solutions of type $(A,B)$ involving Kružkov-type entropy inequalities that can be adapted to any fixed connection $(A,B)$. It is proved that these entropy inequalities imply the $L^1$ contraction property for $L^\infty$ solutions, in contrast to the “piecewise smooth” setting of Adimurthi, Mishra, and Veerappa Gowda. For a fixed connection, these entropy inequalities include a single adapted entropy of the type used by Audusse and Perthame [Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), pp. 253–265]. We prove convergence of a new difference scheme that approximates entropy solutions of type $(A,B)$ for any connection $(A,B)$ if a few parameters are varied. The scheme relies on a modification of the standard Engquist–Osher flux, is simple as no $2\times2$ Riemann solver is involved, and is designed such that the steady-state solution connecting A to B is preserved. In contrast to most analyses of similar problems, our convergence proof is not based on the singular mapping or compensated compactness methods, but on standard spatial variation estimates away from the flux discontinuity. Some numerical examples are presented.