Regularized Nonlinear Solvers for IMEX Methods Applied to Diffusively Corrected Multispecies Kinematic Flow Models

Abstract

Implicit-explicit (IMEX) methods are suitable for the solution of nonlinear convection-diffusion equations, 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 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 multispecies kinematic flow models with strongly degenerate diffusive corrections requires the solution of highly nonlinear and nonsmooth 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 multiclass traffic flow confirm the efficiency of the methods proposed.