## Preprint 2015-29

## Raimund Bürger, Pep Mulet, Lihki Rubio:

### Polynomial viscosity methods for multispecies kinematic flow models

### Abstract:

Multispecies kinematic flow models are dened by systems of strongly coupled, nonlinear first-order conservation laws. They arise in various applications including sedimentation of polydisperse suspensions and multiclass vehicular trafficc. Their numerical approximation is a challenge since the eigenvalues and eigenvectors of the corresponding ux Jacobian matrix have no closed algebraic form. It is demostrated that a recently introduced class of fast first-order finite volume solvers, called PVM (polynomial viscosity matrix) methods [M.J. Castro Daz and E. Fernandez-Nieto, SIAM J Sci Comput 34 (2012), A2173{A2196] can be adapted to multispecies kinematic flows. PVM methods have the advantage that they only need some information about the eigenvalues of the flux Jacobian, and no spectral decomposition of a Roe matrix is needed. In fact, the so-called interlacing property (of eigenvalues with known velocity functions), which holds for several important multispecies kinematic flow models, provides sufficient information for the implementation of PVM methods. Several variants of PVM methods (differing in polynomial degree and the underlying quadrature formula to approximate the Roe matrix) are compared by numerical experiments. It turns out that PVM methods are competitive in accuracy and efficiency with several existing methods, including the HLL method and a spectral WENO scheme that is based on the same interlacing property.

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

**Raimund BüRGER, Pep MULET, Lihki RUBIO: ***Polynomial viscosity methods for multispecies kinematic flow models*. Numerical Methods for Partial Differential Equations, vol. 32, 4, pp. 1265-1288, (2016).