Second-order schemes for conservation laws with discontinuous flux modelling clarifier-thickener units.

*(English)*Zbl 1204.65101The authors consider mainly a staggered first-order difference scheme and its second-order upgrading for a hyperbolic equation in one spatial variable modelling sediment suspension behaviour. The model contains a discontinuous flux, and in former works of the same authors [Comput. Vis. Sci. 6, No. 2–3, 83–91 (2004; Zbl 1299.76283)] the convergence of the first-order scheme to an entropy solution has been considered. Here they upgrade the scheme to (formal) second order by using corresponding correction terms and two kinds of limiters, a simple minmod and a new one (leading to a flux-total variation diminishing scheme), including here a correction for steady sonic rarefactions, too. For the nonlocal limiter proposed, an algorithm is formulated, its properties investigated and its numerical behaviour illustrated by a number of experiments and compared with the simpler scheme. Finally, an extended model including degenerate diffusion is considered also and solved by operator splitting (combining the former hyperbolic scheme and Crank-Nicolson), exhibiting good numerical results for the discontinuous solutions.

Reviewer: Gisbert Stoyan (Budapest)

##### MSC:

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

##### Keywords:

conservation law; discontinuous flux; first-order difference scheme; second-order correction; limiters; numerical experiments; sediment suspension; convergence; total variation diminishing scheme; algorithm; operator splitting; Crank-Nicolson; discontinuous solutions
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\textit{R. Bürger} et al., Numer. Math. 116, No. 4, 579--617 (2010; Zbl 1204.65101)

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##### References:

[1] | Barton N.G., Li C.-H., Spencer S.J.: Control of a surface of discontinuity in continuous thickeners. J. Austral. Math. Soc. Ser. B 33, 269–289 (1992) · Zbl 0758.35009 |

[2] | Bouchut F.: Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources. Frontiers in Mathematics. Birkhäuser Verlag, Basel (2004) · Zbl 1086.65091 |

[3] | Bürger R., García A., Karlsen K.H., Towers J.D.: A family of numerical schemes for kinematic flows with discontinuous flux. J. Eng. Math. 60, 387–425 (2008) · Zbl 1200.76126 |

[4] | Bürger R., Karlsen K.H., Klingenberg C., Risebro N.H.: A front tracking approach to a model of continuous sedimentation in ideal clarifier–thickener units. Nonlin. Anal. Real World Appl. 4, 457–481 (2003) · Zbl 1013.35052 |

[5] | Bürger R., Karlsen K.H., Risebro N.H.: A relaxation scheme for continuous sedimentation in ideal clarifier–thickener units. Comput. Math. Appl. 50, 993–1009 (2005) · Zbl 1122.76063 |

[6] | Bürger R., Karlsen K.H., Risebro N.H., Towers J.D.: Monotone difference approximations for the simulation of clarifier–thickener units. Comput. Visual. Sci. 6, 83–91 (2004) · Zbl 1299.76283 |

[7] | Bürger R., Karlsen K.H., Risebro N.H., Towers J.D.: Well-posedness in BV t and convergence of a difference scheme for continuous sedimentation in ideal clarifier–thickener units. Numer. Math. 97, 25–65 (2004) · Zbl 1053.76047 |

[8] | Bürger R., Karlsen K.H., Towers J.D.: A mathematical model of continuous sedimentation of flocculated suspensions in clarifier–thickener units. SIAM J. Appl. Math. 65, 882–940 (2005) · Zbl 1089.76061 |

[9] | Bürger R., Karlsen K.H., Towers J.D.: An Engquist-Osher type scheme for conservation laws with discontinuous flux adapted to flux connections. SIAM J. Numer. Anal. 47, 1684–1712 (2009) · Zbl 1201.35022 |

[10] | Chancelier J.P., Cohen de Lara M., Pacard F.: Analysis of a conservation PDE with discontinuous flux: a model of settler. SIAM J. Appl. Math. 54, 954–995 (1994) · Zbl 0811.35077 |

[11] | Diehl S.: Dynamic and steady-state behaviour of continuous sedimentation. SIAM J. Appl. Math. 57, 991–1018 (1997) · Zbl 0889.35062 |

[12] | Diehl S.: Operating charts for continuous sedimentation II: step responses. J. Eng. Math. 53, 139–185 (2005) · Zbl 1086.76069 |

[13] | Diehl S.: A uniqueness condition for nonlinear convection-diffusion equations with discontinuous coefficients. J. Hyperbolic Differ. Equ. 6, 127–159 (2009) · Zbl 1180.35305 |

[14] | Engquist B., Osher S.: One-sided difference approximations for nonlinear conservation laws. Math. Comp. 36, 321–351 (1981) · Zbl 0469.65067 |

[15] | Godlewski E., Raviart P.: Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer, New York (1996) · Zbl 0860.65075 |

[16] | Gottlieb S., Shu C., Tadmor E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43, 89–112 (2001) · Zbl 0967.65098 |

[17] | Harten A.: High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49, 357–393 (1983) · Zbl 0565.65050 |

[18] | Kynch G.J.: A theory of sedimentation. Trans. Farad. Soc. 48, 166–176 (1952) · Zbl 0048.22902 |

[19] | Lev O., Rubin E., Sheintuch M.: Steady state analysis of a continuous clarifier–thickener system. AIChE J. 32, 1516–1525 (1986) |

[20] | LeVeque R.J.: Finite Difference Methods for Ordinary and Partial Differential Equations. SIAM, Philadelphia (2007) · Zbl 1127.65080 |

[21] | Osher S., Chakravarthy S.: High resolution schemes and the entropy condition. SIAM J. Numer. Anal. 21, 955–984 (1984) · Zbl 0556.65074 |

[22] | Strang G.: On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5, 506–517 (1968) · Zbl 0184.38503 |

[23] | Sweby P.K.: High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal. 21, 995–1011 (1984) · Zbl 0565.65048 |

[24] | Temple B.: Global solution of the Cauchy problem for a class of 2 {\(\times\)} 2 nonstrictly hyperbolic conservation laws. Adv. Appl. Math. 3, 335–375 (1982) · Zbl 0508.76107 |

[25] | Towers J.D.: A difference scheme for conservation laws with a discontinuous flux: the nonconvex case. SIAM J. Numer. Anal. 39, 1197–1218 (2001) · Zbl 1055.65104 |

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