Linearly Implicit IMEX Runge--Kutta Methods for a Class of Degenerate Convection-Diffusion Problems

Abstract

Multispecies kinematic flow models with strongly degenerate diffusive corrections give rise to systems of nonlinear convection-diffusion equations of arbitrary size. Applications of these systems include models of polydisperse sedimentation and multiclass traffic flow. Implicit-explicit (IMEX) Runge--Kutta (RK) methods are suitable for the solution of these convection-diffusion problems since the stability restrictions, 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 RK scheme for the time integration of the convective part with a diagonally implicit one for the diffusive part. In [R. Burger, P. Mulet, and L. M. Villada, SIAM J. Sci. Comput., 35 (2013), pp. B751--B777] a scheme of this type is proposed, where the nonlinear and nonsmooth systems of algebraic equations arising in the implicit treatment of the degenerate diffusive part are solved by smoothing o...