## Preprint 2012-21

## Raimund Bürger, Pep Mulet, Luis M. Villada:

### Implicit-explicit methods for diffusively corrected multi-species kinematic flow models

### Abstract:

Implicit-explicit methods are a suitable choice for the solution of nonlinear convection-diffusion equations, since the stability estrictions, coming from the explicitly treated convective part, are much less severe than those that would be deduced from an explicit treatment of the diffusive term. These schemes usually combine an explicit Runge-Kutta scheme for the time integration of the convective part with a diagonally implicit one for the diffusive part. The application of these schemes to multi-species kinematic flow models with strongly degenerate diffusive corrections requires the solution of highly nonlinear and non-smooth systems of algebraic equations. Since the efficient solution of these systems by the Newton-Raphson method requires some degree of smoothness, it is proposed to regularize the diffusion coefficients in the model and to apply suitable techniques to solve these nonlinear systems in an efficient way. Numerical examples arising from models of polydisperse sedimentation and multi-class traffic flow confirm the efficiency of the methods proposed.

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

**Raimund BüRGER, Pep MULET, Luis M. VILLADA: ***Regularized nonlinear solvers for IMEX methods applied to diffusively corrected multi-species kinematic flow models*. SIAM Journal on Scientific Computing, 35 (3), pp. B751-B777, (2013).