On diffusively corrected multispecies kinematic flow models

On diffusively corrected multispecies kinematic flow models∗ Raimund B¨urger† Abstract This presentation provides a survey of some recent results related to efficient numerical methods for the numerical solution of convection-diffusion systems of the form ∂Φ ∂t +∂ ∂xf(Φ) = ∂ ∂xB(Φ)∂Φ ∂x , x ∈Ω⊂R, t ≥0,(1) where tis time, x∈Ω is spatial position where Ω is a bounded interval, Φ = (φ1, . . . , φN)T is the vector of unknowns that usually denote the partial density or concentration of one of the Nspecies that make up the disperse phase, f(Φ) = (f1(Φ), . . . , fN(Φ))Tis a flux vector where fj(Φ) = φjvj(Φ) for specified velocity functions vj(Φ), and Bis matrix that expresses a diffusive correction of the first-order model ∂Φ/∂x +∂f(Φ)/∂x =0. Models of this kind arise as one-dimensional models of the flow of one (disperse) substance through a continuous fluid. Applications include the settling of polydisperse suspensions of solid particles in a viscous fluid [1, 2, 3, 6], multiclass vehicular traffic under the effect of anticipation distances and reaction times [4, 5], the settling of dipersions and emulsions [7], and chromatography [8]. In many of these applications it is assumed that B(Φ) = 0on a Φ-set of positive N-dimensional measure, so that (1) becomes strongly degenerate, and moreover Bmay depend discontinuously on Φ. For the numerical solution of (1) along with initial and boundary conditions that depend on the appliction under study these properties pose a number of difficulties whose partial solution will be addressed. For instance, it is well known that implicit-explicit (IMEX) numerical scheme that are based on discretizing the convective and diffusive parts of (1) are a potentially suitable tool to avoid the severe time step limitation associated with fully explicit discretization of (1). However, their implementation relies on the efficient numerical solution of the nonlinear systems of algebraic equations arising from the discretization of (1), which can not be achieved by standard Newton-Raphson techniques when Bdepends discontinuously on Φ. A combined smoothing and line search technique [5] solves the problem of solving the corresponding nonlinearly implicit equations. Alternatively, this problem can be avoided by the construction of so-called linearly implicit methods [2] that are slightly less accurate, but noticeably more efficient than their nonlinearly implicit counterparts. The main collaborators in this research are Pep Mulet (Universitat de Val`encia, Spain) and Luis Miguel Villada (Universidad del B´ıo-B´ıo, Concepci´on). ∗This work was partially supported by CONICYT-Chile through BASAL project CMM, Universidad de Chile; Fondecyt project 1170473; CRHIAM, project CONICYT/FONDAP/15130015; and by Centro de Investigaci´on en Ingenier´ıa Matem´atica (CI2MA), Universidad de Concepci´on. †CI2MA and Departamento de Ingenier´ıa Matem´atica, Universidad de Concepci´on, Casilla 160-C, Concepci´on, Chile, email: [email protected] 1 References [1] S. Berres, R. B¨urger, K.H. Karlsen and E.M. Tory, ‘Strongly degenerate parabolichyperbolic systems modeling polydisperse sedimentation with compression’, SIAM J. Appl. Math. 64 (2003), 41–80. [2] S. Boscarino, R. B¨urger, P. Mulet, G. Russo and L.M. Villada, ‘Linearly implicit IMEX Runge-Kutta Methods for a class of degenerate convection-diffusion problems’, SIAM J. Sci. Comput. 37 (2015), B305–B331. [3] S. Boscarino, R. B¨urger, P. Mulet, G. Russo and L.M. Villada, ‘On linearly implicit IMEX Runge-Kutta Methods for degenerate convection-diffusion problems modelling polydisperse sedimentation’, Bull. Braz. Math. Soc. (N. S.) 47 (2016), 171–185. [4] R. B¨urger, P. Mulet and L.M. Villada, ‘A diffusively corrected multiclass LighthillWhitham-Richards traffic model with anticipation lengths and reaction times’, Adv. Appl. Math. Mech. 5(2013), 728–758. [5] R. B¨urger, P. Mulet and L.M. Villada, ‘Regularized nonlinear solvers for IMEX methods applied to diffusively corrected multi-species kinematic flow models’, SIAM J. Sci. Comput. 35 (2013), B751–B777. [6] R. B¨urger, S. Diehl, M.C. Mart´ı, P. Mulet, I. Nopens, E. Torfs and P.A. Vanrolleghem, ‘Numerical solution of a multi-class model for batch settling in water resource recovery facilities’; submitted. [7] R. B¨urger, P. Mulet and L. Rubio, ‘Implicit-explicit methods for the efficient simulation of the settling of dispersions of droplets and colloidal particles’; submitted. [8] R. B¨urger, P. Mulet, L. Rubio and M. Sep´ulveda, ’Linearly implicit IMEX schemes for the equilibrium dispersive model of chromatography’. Preprint 2017-11, Centro de Investigaci´on en Ingenier´ıa Matem´atica; submitted. 2