## CONVERGENCE OF GODUNOV TYPE METHODS FOR A CONSERVATION LAW WITH A SPATIALLY VARYING DISCONTINUOUS FLUX FUNCTION

Citations: | 5 - 1 self |

### BibTeX

@MISC{Mishra_convergenceof,

author = {Siddhartha Mishra and G. D. Veerappa Gowda},

title = {CONVERGENCE OF GODUNOV TYPE METHODS FOR A CONSERVATION LAW WITH A SPATIALLY VARYING DISCONTINUOUS FLUX FUNCTION},

year = {}

}

### OpenURL

### Abstract

Abstract. We deal with single conservation laws with a spatially varying and possibly discontinuous coefficient. This equation includes as a special case single conservation laws with conservative and possibly singular source terms. We extend the framework of optimal entropy solutions for these classes of equations based on a two-step approach. In the first step, an interface connection vector is used to define infinite classes of entropy solutions. We show that each of these classes of solutions is stable in L1. This allows for the possibility of choosing one of these classes of solutions based on the physics of the problem. In the second step, we define optimal entropy solutions based on the solution of a certain optimization problem at the discontinuities of the coefficient. This method leads to optimal entropy solutions that are consistent with physically observed solutions in two-phase flows in heterogeneous porous media. Another central aim of this paper is to develop suitable numerical schemes for these equations. We develop and analyze a set of Godunov type finite volume methods that are based on exact solutions of the corresponding Riemann problem. Numerical experiments are shown comparing the performance of these schemes on a set of test problems. 1.

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Citation Context ...In this section, we describe two numerical schemes of the Godunov finite volume type for (1). In order to do so, we start with some definitions. Let h ∈ Lip[s, S]. Then the standard Godunov flux (see =-=[13]-=-) is given by ⎧ ⎨ min h(θ) if a ≤ b, θ∈[a,b] (15) H(a, b) = ⎩ max h(θ) if a ≥ b. θ∈[b,a] Let h − ,h + ∈ [s, S] be two functions such that they have exactly one minimum and no maxima in [s, S], and let... |

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Citation Context ...ation of coefficients as in [18], by front tracking as in [12, 17, 18], by explicit Hopf-Lax formulas in [1] and by proving convergence of numerical schemes of the Godunov or Enquist-Osher type as in =-=[2, 25, 27, 28, 19, 6]-=- and the Lax-Friedrichs type as in [21]. Similarly the case of two-phase flow in a medium with changing rock types has been considered in [22] and [15]. The applications relating to the clarifier thic... |

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Citation Context ...er. Equation (1) has been widely studied from both the theoretical and numerical points of view in recent years. Entropy conditions for some specific cases of (1) were devised by Gimse and Risebro in =-=[12]-=-, Diehl in [8, 9], Klingenberg and Risebro in [17] and for the full equation by Karlsen, Risebro and Towers in [20]. They used a modified Kruzkhov type entropy condition and showed L1 stability of ent... |

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Citation Context ... [12, 17, 18], by explicit Hopf-Lax formulas in [1] and by proving convergence of numerical schemes of the Godunov or Enquist-Osher type as in [2, 25, 27, 28, 19, 6] and the Lax-Friedrichs type as in =-=[21]-=-. Similarly the case of two-phase flow in a medium with changing rock types has been considered in [22] and [15]. The applications relating to the clarifier thickener unit have been considered in [8, ... |

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Citation Context ...ation of coefficients as in [18], by front tracking as in [12, 17, 18], by explicit Hopf-Lax formulas in [1] and by proving convergence of numerical schemes of the Godunov or Enquist-Osher type as in =-=[2, 25, 27, 28, 19, 6]-=- and the Lax-Friedrichs type as in [21]. Similarly the case of two-phase flow in a medium with changing rock types has been considered in [22] and [15]. The applications relating to the clarifier thic... |

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Citation Context ... aligned discretization type. For the particular case of conservative source terms (and convex fluxes), an aligned discretization based Godunov type scheme was developed for (2) by Greenberg et al in =-=[10]-=-. This scheme was shown to work very well on test cases in [10] and was reported to be better than staggered grid schemes in [28]. A proof of convergence of this scheme will be of independent interest... |

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Citation Context ...type entropy condition and showed L1 stability of entropy solutions. Concurrently, several existence results for the entropy solutions have been obtained by using regularization of coefficients as in =-=[18]-=-, by front tracking as in [12, 17, 18], by explicit Hopf-Lax formulas in [1] and by proving convergence of numerical schemes of the Godunov or Enquist-Osher type as in [2, 25, 27, 28, 19, 6] and the L... |

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Citation Context ...ation of coefficients as in [18], by front tracking as in [12, 17, 18], by explicit Hopf-Lax formulas in [1] and by proving convergence of numerical schemes of the Godunov or Enquist-Osher type as in =-=[2, 25, 27, 28, 19, 6]-=- and the Lax-Friedrichs type as in [21]. Similarly the case of two-phase flow in a medium with changing rock types has been considered in [22] and [15]. The applications relating to the clarifier thic... |

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Citation Context ...etails), in modeling gravity settling in an ideal clarifier thickener unit used in waste water treatment plants (see [6]), in the modeling of traffic on highways with changing surface conditions (see =-=[23]-=-) and in ion etching in the semiconductor industry (see [24]). A detailed account of various applications is found in [26]. Received by the editor September 19, 2005 and, in revised form, June 23, 200... |

12 |
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Citation Context ...ntly, several existence results for the entropy solutions have been obtained by using regularization of coefficients as in [18], by front tracking as in [12, 17, 18], by explicit Hopf-Lax formulas in =-=[1]-=- and by proving convergence of numerical schemes of the Godunov or Enquist-Osher type as in [2, 25, 27, 28, 19, 6] and the Lax-Friedrichs type as in [21]. Similarly the case of two-phase flow in a med... |

9 |
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Citation Context ...the Godunov or Enquist-Osher type as in [2, 25, 27, 28, 19, 6] and the Lax-Friedrichs type as in [21]. Similarly the case of two-phase flow in a medium with changing rock types has been considered in =-=[22]-=- and [15]. The applications relating to the clarifier thickener unit have been considered in [8, 6], etc. In a recent series of papers [3, 4, 5], the authors have embarked on a program to formulate a ... |

6 |
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Citation Context ...thickener unit used in waste water treatment plants (see [6]), in the modeling of traffic on highways with changing surface conditions (see [23]) and in ion etching in the semiconductor industry (see =-=[24]-=-). A detailed account of various applications is found in [26]. Received by the editor September 19, 2005 and, in revised form, June 23, 2006. 2000 Mathematics Subject Classification. Primary 35L65, 6... |

4 |
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Citation Context ..., f is the flux function and k is a spatially varying, possibly discontinuous coefficient. The discontinuous coefficient leads to the fact that neither the standard well-posedness theory of Kruzkhov (=-=[16]-=-) nor the usual numerical methods (see [14]) apply in this case. Equations of the above type arise while dealing with fluid flows in heterogeneous media such as in two phase flow in a porous medium wi... |

4 |
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Citation Context ... in the modeling of traffic on highways with changing surface conditions (see [23]) and in ion etching in the semiconductor industry (see [24]). A detailed account of various applications is found in =-=[26]-=-. Received by the editor September 19, 2005 and, in revised form, June 23, 2006. 2000 Mathematics Subject Classification. Primary 35L65, 65M06, 65M12. 1219 c○2007 American Mathematical Society Reverts... |

2 |
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1 |
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Citation Context ... in a medium with changing rock types has been considered in [22] and [15]. The applications relating to the clarifier thickener unit have been considered in [8, 6], etc. In a recent series of papers =-=[3, 4, 5]-=-, the authors have embarked on a program to formulate a proper notion of entropy solutions for (1) and show their existence and uniqueness. In [3], we proposed a new concept of entropy solutions, name... |

1 |
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Citation Context ...Equations of the above type arise while dealing with fluid flows in heterogeneous media such as in two phase flow in a porous medium with changing rock types that arise in the petroleum industry (see =-=[15]-=- for more details), in modeling gravity settling in an ideal clarifier thickener unit used in waste water treatment plants (see [6]), in the modeling of traffic on highways with changing surface condi... |