On the hyperbolicity of certain models of polydisperse sedimentation

Bürger, Raimund; Donat, Rosa; Mulet, Pep; Vega, Carlos A.

doi:10.1002/mma.1617

Cited by 3 publications

(6 citation statements)

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“…In this case,

{scriptJ}_{bold-italicf} (Φ)

becomes a rank‐

m

perturbation of a diagonal matrix. This property has made it possible to estimate the hyperbolicity region for a number of polydisperse sedimentation models with

m \leq 4

(see ) by the so‐called secular equation . The hyperbolicity and the interlacing of the (unknown) eigenvalues of

{scriptJ}_{bold-italicf} (Φ)

with the (known) velocities

v_{1}, \dots, v_{N}

form the key ingredients for the construction of efficient characteristic‐wise (spectral) weighted essentially non‐oscillatory (WENO) schemes for , denoted “WENO‐SPEC‐INT” according to , which are employed herein to generate reference solutions to assess the performance of L‐AR schemes for the MLB model of polydisperse sedimentation.…”

Section: Preliminariesmentioning

confidence: 99%

“…For coefficients

β_{0}, β_{1}, β_{2} < 0

β_{3} = 0

and

β_{0}, \dots, β_{3} < 0

, the DG model with coefficients

S_{i j}

defined by corresponds to the case

m = 3

and

m = 4

of , respectively, as in each case by , the formula for

v_{i}

depends on

m

independent linear combinations of

ϕ_{1}, \dots, ϕ_{N}

. For general

N

it has been possible to estimate the hyperbolicity region of the DG model in terms of

ϕ_{\max}

and the width of the particle size distribution expressed by the smallest particle size ratio

δ_{N}^{1 / 2} = d_{N} / d_{1}

. The approximate hyperbolicity analysis of the DG model, conducted in by the secular equation approach , is outside the scope of the paper; however, in the numerical examples the parameters of the DG model have been chosen in such a way that hyperbolicity is ensured.…”

Section: Preliminariesmentioning

confidence: 99%

“…For general

N

it has been possible to estimate the hyperbolicity region of the DG model in terms of

ϕ_{\max}

and the width of the particle size distribution expressed by the smallest particle size ratio

δ_{N}^{1 / 2} = d_{N} / d_{1}

Section: Preliminariesmentioning

confidence: 99%

“…In this case, J f ( ) becomes a rankm perturbation of a diagonal matrix. This property has made it possible to estimate the hyperbolicity region for a number of polydisperse sedimentation models with m ≤ 4 (see [24,27]) by the so-called secular equation [28]. The hyperbolicity and the interlacing of the (unknown) eigenvalues of J f ( ) with the (known) velocities v 1 , .…”

Section: A Models Of Polydisperse Sedimentationmentioning

confidence: 99%

“…, φ N [24]. For general N it has been possible to estimate the hyperbolicity region of the DG model [27] in terms of φ max and the width of the particle size distribution expressed by the smallest particle size ratio…”

Section: )mentioning

confidence: 99%

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Antidiffusive Lagrangian‐remap schemes for models of polydisperse sedimentation

Bürger

Chalons

Villada

2015

Numerical Methods Partial

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One-dimensional models of gravity-driven sedimentation of polydisperse suspensions with particles that belong to N size classes give rise to systems of N strongly coupled, nonlinear first-order conservation laws for the local solids volume fractions. As the eigenvalues and eigenvectors of the flux Jacobian have no closed algebraic form, characteristic-wise numerical schemes for these models become involved. Alternative simple schemes for this model directly utilize the velocity functions and are based on splitting the system of conservation laws into two different first-order quasi-linear systems, which are solved successively for each time iteration, namely, the Lagrangian and remap steps (so-called Lagrangian-remap [LR] schemes). This approach was advanced in (Bürger, Chalons, and Villada, SIAM J Sci Comput 35 (2013), B1341-B1368) for a multiclass Lighthill-Whitham-Richards traffic model with nonnegative velocities. By incorporating recent antidiffusive techniques for transport equations a new version of these Lagrangian-antidiffusive remap (L-AR) schemes for the polydisperse sedimentation model is constructed. These L-AR schemes are supported by a partial analysis for N = 1. They are total variation diminishing under a suitable CFL condition and therefore converge to a weak solution. Numerical examples illustrate that these schemes, including a more accurate version based on MUSCL extrapolation, are competitive in accuracy and efficiency with several existing schemes.

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“…In this case,

{scriptJ}_{bold-italicf} (Φ)

becomes a rank‐

m

perturbation of a diagonal matrix. This property has made it possible to estimate the hyperbolicity region for a number of polydisperse sedimentation models with