Central schemes and systems of conservation laws with discontinuous coefficients modeling gravity separation of polydisperse suspensions

摘要

In this paper, two existing one-dimensional mathematical models, one for continuous sedimentation of monodisperse suspensions and one for settling of polydisperse suspensions, are combined into a model of continuous separation of polydisperse mixtures. This model can be written as a first-order system of conservation laws for the local concentrations of each particle species with a flux vector that depends discontinuously on the space variable. This application motivates the extension of the Kurganov–Tadmor central difference scheme to systems with discontinuous flux. The new central schemes are based on discretizing an enlarged system in which the discontinuous coefficients are viewed as additional conservation laws. These additional conservation laws can either be discretized and the evolution of the discontinuity parameters is calculated in each time step, or solved exactly, that is, the discontinuity parameters are kept constant (with respect to time). Numerical examples and an L1 error study show that the Kurganov–Tadmor scheme with first-order in time discretization produces spurious oscillations, whereas its semi-discrete version, discretized by a second-order Runge–Kutta scheme, produces good results. The scheme with discontinuity parameters kept constant is slightly more accurate than when these are evolved. Numerical examples illustrate the application to separation of polydisperse suspensions.