## Preprint 2015-04

## Raimund Bürger, Christophe Chalons, Luis M. Villada:

### Antidiffusive Lagrangian-remap schemes for models of polydisperse sedimentation

### Abstract:

Models of sedimentation of polydisperse suspensions of small particles in a viscous fluid that belong to N size classes give rise to systems of N strongly coupled, nonlinear first-order conservation laws for the local solids volume fractions as functions of depth and time. The settling velocities usually have variable sign depending on local fluctuations of the density of the mixture, and the model is posed with zero-flux boundary conditions for batch settling in a column. Since the eigenvalues and eigenvectors of the flux Jacobian have no closed algebraic form, characteristic-wise numerical schemes for these models become involved. Alternative simple schemes for this model directly utilize the velocity functions and are based on splitting the system of conservation laws into two different first-order quasi-linear systems, which are solved successively for each time iteration, namely, the Lagrangian and remap steps (so-called Lagrangian-remap (LR) schemes). This approach was advanced in [R. Bürger, C. Chalons and L.M. Villada, SIAM J. Sci. Comput. 35 (2013) B1341–B1368] for a multiclass Lighthill-Whitham-Richards traffic model with nonnegative velocities. By incorporating recent antidiffusive techniques for transport equations a new version of these Lagrangian-antidiffusive remap (L-AR) schemes for the polydisperse sedimentation model is constructed. These L-AR schemes are supported by a partial analysis for N = 1. They are total variation diminishing under a suitable CFL condition and therefore converge to a weak solution. Numerical examples for several values of N illustrate that these schemes, including a more accurate version based on MUSCL extrapolation, are competitive in accuracy and efficiency with several existing schemes.

This preprint gave rise to the following definitive publication(s):

**Raimund BüRGER, Christophe CHALONS, Luis M. VILLADA: ***Antidiffusive Lagrangian-remap schemes for models of polydisperse sedimentation*. Numerical Methods for Partial Differential Equations, vol. 32, 4, pp. 1109-1136, (2016).