Implicit-explicit methods for a class of nonlinear nonlocal gradient flow equations modelling collective behaviour

Abstract The numerical solution of nonlinear convection-diffusion equations with nonlocal flux by explicit finite difference methods is costly due to the local spatial convolution within the convective numerical flux and the disadvantageous Courant-Friedrichs-Lewy (CFL) condition caused by the diffusion term. More efficient numerical methods are obtained by applying second-order implicit-explicit (IMEX) Runge-Kutta time discretizations to an available explicit scheme for such models in Carrillo et al. (2015) [13] . The resulting IMEX-RK methods require solving nonlinear algebraic systems in every time step. It is proven, for a general number of space dimensions, that this method is well def…

Spectral WENO schemes with Adaptive Mesh Refinement for models of polydisperse sedimentation

The sedimentation of a polydisperse suspension with particles belonging to N size classes (species) can be described by a system of N nonlinear, strongly coupled scalar first-order conservation laws. Its solutions usually exhibit kinematic shocks separating areas of different composition. Based on the so-called secular equation [J. Anderson, Lin. Alg. Appl. 246, 49–70 (1996)], which provides access to the spectral decomposition of the Jacobian of the flux vector for this class of models, Burger et al. [J. Comput. Phys. 230, 2322–2344 (2011)] proposed a spectral weighted essentially non-oscillatory (WENO) scheme for the numerical solution of the model. It is demonstrated that the efficiency …

Implicit–explicit schemes for nonlinear nonlocal equations with a gradient flow structure in one space dimension

Modelling the spatial-temporal progression of the 2009 A/H1N1 influenza pandemic in Chile

A spatial-temporal transmission model of 2009 A/H1N1 pandemic influenza across Chile, a country that spans a large latitudinal range, is developed to characterize the spatial variation in peak timing of that pandemic as a function of local transmission rates, spatial connectivity assumptions for Chilean regions, and the putative location of introduction of the novel virus into the country. Specifically, a metapopulation SEIR (susceptible-exposed-infected-removed) compartmental model that tracks the transmission dynamics of influenza in 15 Chilean regions is calibrated. The model incorporates population mobility among neighboring regions and indirect mobility to and from other regions via th…

Numerical solution of a spatio-temporal gender-structured model for hantavirus infection in rodents.

In this article we describe the transmission dynamics of hantavirus in rodents using a spatio-temporal susceptible-exposed-infective-recovered (SEIR) compartmental model that distinguishes between male and female subpopulations [L.J.S. Allen, R.K. McCormack and C.B. Jonsson, Bull. Math. Biol. 68 (2006), 511--524]. Both subpopulations are assumed to differ in their movement with respect to local variations in the densities of their own and the opposite gender group. Three alternative models for the movement of the male individuals are examined. In some cases the movement is not only directed by the gradient of a density (as in the standard diffusive case), but also by a non-local convolution…