Efficient implementation of essentially non-oscillatory shock-capturing schemes
Journal of Computational Physics
Weighted essentially non-oscillatory schemes
Journal of Computational Physics
Efficient implementation of weighted ENO schemes
Journal of Computational Physics
Total variation diminishing Runge-Kutta schemes
Mathematics of Computation
Two Barriers on Strong-Stability-Preserving Time Discretization Methods
Journal of Scientific Computing
Journal of Computational Physics
Mapped weighted essentially non-oscillatory schemes: Achieving optimal order near critical points
Journal of Computational Physics
Journal of Computational Physics
Adaptive multiresolution WENO schemes for multi-species kinematic flow models
Journal of Computational Physics
Characteristic-Based Schemes for Multi-Class Lighthill-Whitham-Richards Traffic Models
Journal of Scientific Computing
High Order Strong Stability Preserving Time Discretizations
Journal of Scientific Computing
A multilayer shallow water system for polydisperse sedimentation
Journal of Computational Physics
Hi-index | 31.45 |
The sedimentation of a polydisperse suspension of small rigid spheres of the same density, but which belong to a finite number of species (size classes), can be described by a spatially one-dimensional system of first-order, nonlinear, strongly coupled conservation laws. The unknowns are the volume fractions (concentrations) of each species as functions of depth and time. Typical solutions, e.g. for batch settling in a column, include discontinuities (kinematic shocks) separating areas of different composition. The accurate numerical approximation of these solutions is a challenge since closed-form eigenvalues and eigenvectors of the flux Jacobian are usually not available, and the characteristic fields are neither genuinely nonlinear nor linearly degenerate. However, the flux vectors associated with the widely used models by Masliyah, Lockett and Bassoon (MLB model) and Hofler and Schwarzer (HS model) give rise to Jacobians that are low-rank perturbations of a diagonal matrix. This property allows to apply a convenient hyperbolicity criterion that has become known as the ''secular equation'' [J. Anderson, A secular equation for the eigenvalues of a diagonal matrix perturbation, Lin. Alg. Appl. 246 (1996) 49-70]. This criterion was recently applied [R. Burger, R. Donat, P. Mulet, C.A. Vega, Hyperbolicity analysis of polydisperse sedimentation models via a secular equation for the flux Jacobian, SIAM J. Appl. Math. 70 (2010) 2186-2213] to prove that the MLB and HS models are strictly hyperbolic under easily verifiable conditions, that their eigenvalues interlace with the velocities of the species that form the flux vector (so the velocities are good starting values for a root finder), and that the corresponding eigenvectors can be calculated with acceptable effort. In the present work, the newly available characteristic information is exploited for the implementation of characteristic-wise (spectral) weighted essentially non-oscillatory (WENO) schemes for the MLB and HS models. Numerical examples illustrate that WENO schemes which use this spectral information are superior in resolution, and even in efficiency for the same overall resolution, to component-wise WENO schemes.