scholarly journals Convergence Rates of Monotone Schemes for Conservation Laws with Discontinuous Flux

2020 ◽  
Vol 58 (1) ◽  
pp. 607-629
Author(s):  
Jayesh Badwaik ◽  
Adrian M. Ruf
Keyword(s):  
Conservation Laws ◽  
Convergence Rates ◽  
Monotone Schemes ◽  
Discontinuous Flux

scholarly journals Flux-stability for conservation laws with discontinuous flux and convergence rates of the front tracking method

2021 ◽  
Author(s):  
Adrian M Ruf
Keyword(s):  
Convergence Rate ◽  
Conservation Laws ◽  
Convergence Rates ◽  
Numerical Solutions ◽  
Stability Result ◽  
Front Tracking ◽  
Tracking Method ◽  
Front Tracking Method ◽  
Discontinuous Flux ◽  
Scalar Conservation

Abstract We prove that adapted entropy solutions of scalar conservation laws with discontinuous flux are stable with respect to changes in the flux under the assumption that the flux is strictly monotone in $u$ and the spatial dependency is piecewise constant with finitely many discontinuities. We use this stability result to prove a convergence rate for the front tracking method—a numerical method that is widely used in the field of conservation laws with discontinuous flux. To the best of our knowledge, both of these results are the first of their kind in the literature on conservation laws with discontinuous flux. We also present numerical experiments verifying the convergence rate results and comparing numerical solutions computed with the front tracking method to finite volume approximations.

scholarly journals A Bhatnagar–Gross–Krook approximation to scalar conservation laws with discontinuous flux

2010 ◽  
Vol 140 (5) ◽  
pp. 953-972 ◽  
Author(s):  
F. Berthelin ◽  
J. Vovelle
Keyword(s):  
Conservation Laws ◽  
Entropy Solution ◽  
Limit Problem ◽  
Scalar Conservation Laws ◽  
First Order ◽  
The Cauchy Problem ◽  
Discontinuous Flux ◽  
Scalar Conservation ◽  
Bgk Approximation ◽  
Well Posed

AbstractWe study the Bhatnagar–Gross–Krook (BGK) approximation to first-order scalar conservation laws with a flux which is discontinuous in the space variable. We show that the Cauchy problem for the BGK approximation is well posed and that, as the relaxation parameter tends to 0, it converges to the (entropy) solution of the limit problem.

A system of conservation laws with discontinuous flux modelling flotation with sedimentation

2019 ◽  
Vol 84 (5) ◽  
pp. 930-973 ◽  
Author(s):  
Raimund Bürger ◽  
Stefan Diehl ◽  
María del Carmen Martí
Keyword(s):  
Conservation Laws ◽  
Steady States ◽  
Solid Particles ◽  
Flotation Column ◽  
Drift Flux ◽  
Hydrophobic Particles ◽  
Discontinuous Flux ◽  
Scalar Conservation ◽  
Steady State Solutions ◽  
Hydrophilic Particles

Abstract The continuous unit operation of flotation is extensively used in mineral processing, wastewater treatment and other applications for selectively separating hydrophobic particles (or droplets) from hydrophilic ones, where both are suspended in a viscous fluid. Within a flotation column, the hydrophobic particles are attached to gas bubbles that are injected and float as aggregates forming a foam or froth at the top that is skimmed. The hydrophilic particles sediment and are discharged at the bottom. The hydrodynamics of a flotation column is described in simplified form by studying three phases, namely the fluid, the aggregates and solid particles, in one space dimension. The relative movements between the phases are given by constitutive drift-flux functions. The resulting model is a system of two scalar conservation laws with a multiply discontinuous flux for the aggregates and solids volume fractions as functions of height and time. The model is of triangular nature since one equation can be solved independently of the other. Based on the theory of conservation laws with discontinuous flux, steady-state solutions that satisfy all jump and entropy conditions are constructed. For the existence of the industrially relevant steady states, conditions on feed flows and concentrations are established and mapped as ‘operating charts’. A numerical method that exploits the triangular structure is formulated on a pair of staggered grids and is employed for the simulation of the fill-up and transitions between steady states of the flotation column.

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