Numerical Methods for Conservation Laws With Discontinuous Coefficients

Mishra, Siddhartha

doi:10.1016/bs.hna.2016.11.002

Cited by 8 publications

(9 citation statements)

References 62 publications

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“…It should be stressed that the first-order spatial derivatives in Eqs. (14)(15) are discretized with opposite one-sided finite differences in the corresponding predictor and corrector stages. The forward differencing operator is used in the predictor step and the backward differencing operator is used in the corrector step.…”

Section: Second-order Schemesmentioning

confidence: 99%

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A Predictor-Corrector Scheme for Conservation Equations with Discontinuous Coefficients

Okhovati

Izadi

2020

J. Math. Fund. Sci.

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In this paper we propose an explicit predictor-corrector finite difference scheme to numerically solve one-dimensional conservation laws with discontinuous flux function appearing in various physical model problems, such as traffic flow and two-phase flow in porous media. The proposed method is based on the second-order MacCormack finite difference scheme and the solution is obtained by correcting first-order schemes. It is shown that the order of convergence is quadratic in the grid spacing for uniform grids when applied to problems with discontinuity. To illustrate some properties of the proposed scheme, numerical results applied to linear as well as non-linear problems are presented.

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Section: Second-order Schemesmentioning

confidence: 99%

“…For an overview of some recent results devoted to problem Eq. (3), we refer the interested reader to Mishra [14] and the references therein. We note that if ( , ( )) is not dependent on , then Eq.…”

Section: Introductionmentioning

confidence: 99%

A Predictor-Corrector Scheme for Conservation Equations with Discontinuous Coefficients

Okhovati

Izadi

2020

J. Math. Fund. Sci.

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“…For instance, applications of (1.3) also include traffic flow with discontinuously changing road surface conditions, ion etching, two-phase flow in heterogeneous porous media, and medical applications (see [5], [24,Ch. 8], and [28] for overviews and references). We use here the admissibility condition from [14], which has proved to be the natural one for the related problem of continuous sedimentation [15].…”

Section: 2mentioning

confidence: 99%

A conservation law with multiply discontinuous flux modelling a flotation column

Bürger

Diehl²,

Martí

2018

Networks &Amp; Heterogeneous Media

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Flotation is a unit operation extensively used in the recovery of valuable minerals in mineral processing and related applications. Essential insight to the hydrodynamics of a flotation column can be obtained by studying just two phases: gas and fluid. To this end, the approach based on the drift-flux theory, proposed in similar form by several authors, is reformulated as a one-dimensional non-linear conservation law with a multiply discontinuous flux. The unknown is the gas volume fraction as a function of height and time, and the flux function depends discontinuously on spatial position due to several feed inlets. The resulting model is similar, but not equivalent, to previously studied clarifier-thickener models for solid-liquid separation and therefore adds a new real-world application to the field of conservation laws with discontinuous flux. Steady-state solutions are studied in detail, including their construction by applying an appropriate entropy condition across each flux discontinuity. This analysis leads to operating charts and tables collecting all possible steady states along with some necessary conditions for their feasibility in each case. Numerical experiments show that the transient model recovers the steady states, depending on the feed rates of the different inlets.

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“…The transmission map approach 4. developed in this paper leads, as for the approaches 1-3, to a complete description of the set of the L 1 -dissipative germs in the bell-shaped case. We demonstrate in this paper that it can be applied directly for the general non-bell-shaped case, while the approach by connections 2. becomes quite technical in the non-bell-shaped case (we refer in particular to the PhD thesis of S. Mishra [61], where this problem has been studied thoroughly).…”

Section: The Case Of Bell-shaped Fluxesmentioning

confidence: 99%

On interface transmission conditions for conservation laws with discontinuous flux of general shape

Andreïanov

Cancès

2015

J. Hyper. Differential Equations

106

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International audienceConservation laws of the form $\partial_t u+ \partial_x f(x;u)=0$ with space-discontinuous flux $f(x;\cdot)=f_l(\cdot)\mathbf{1}_{x<0}+f_r(\cdot)\mathbf{1}_{x>0}$ were deeply investigated in the last ten years, with a particular emphasis in the case where the fluxes are ''bell-shaped". In this paper, we introduce and exploit the idea of transmission maps for the interface condition at the discontinuity, leading to the well-posedness for the Cauchy problem with general shape of $f_{l,r}$. The design and the convergence of monotone Finite Volume schemes based on one-sided approximate Riemann solvers is then assessed. We conclude the paper by illustrating our approach by several examples coming from real-life applications

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Numerical Methods for Conservation Laws With Discontinuous Coefficients

Cited by 8 publications

References 62 publications

A Predictor-Corrector Scheme for Conservation Equations with Discontinuous Coefficients

A Predictor-Corrector Scheme for Conservation Equations with Discontinuous Coefficients

A conservation law with multiply discontinuous flux modelling a flotation column

On interface transmission conditions for conservation laws with discontinuous flux of general shape

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