On the implementation of WENO schemes for a class of polydisperse sedimentation models. (English) Zbl 1391.76465
Summary: 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 Höfler 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, Linear Algebra Appl. 246, 49–70 (1996; Zbl 0861.15006)]. This criterion was recently applied [R. Bürger et al., SIAM J. Appl. Math. 70, No. 7, 2186–2213 (2010; Zbl 1223.35226)] 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.
MSC:
| 76M20 | Finite difference methods applied to problems in fluid mechanics |
| 76T20 | Suspensions |
| 35L65 | Hyperbolic conservation laws |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
WENO schemes; systems of conservation laws; characteristic-wise schemes; spectral decomposition; secular equation; flux splitting; polydisperse sedimentation; multi-species kinematic flow modelReferences:
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