## Convergence of relaxation schemes for hyperbolic conservation laws with stiff source terms (1999)

Venue: | Math. Comput |

Citations: | 7 - 0 self |

### BibTeX

@ARTICLE{Chalabi99convergenceof,

author = {A. Chalabi},

title = {Convergence of relaxation schemes for hyperbolic conservation laws with stiff source terms},

journal = {Math. Comput},

year = {1999},

volume = {68},

pages = {68--955}

}

### OpenURL

### Abstract

Abstract. We focus in this study on the convergence of a class of relaxation numerical schemes for hyperbolic scalar conservation laws including stiff source terms. Following Jin and Xin, we use as approximation of the scalar conservation law, a semi-linear hyperbolic system with a second stiff source term. This allows us to avoid the use of a Riemann solver in the construction of the numerical schemes. The convergence of the approximate solution toward a weak solution is established in the cases of first and second order accurate MUSCL relaxed methods. 1.

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Citation Context ...t the former result was proved without any CFL condition.s964 A. CHALABI 5.3. The MUSCL semi-implicit relaxed scheme. To construct a second order accurate (in space) scheme using the Van-Leer method (=-=[30]-=-), we will need the initial boundary values inside cell j: with (5.9) and (v n + √ au n ) j+1/2 =(v n + √ au n )j +1/2hδ +n j , (v n − √ au n ) j+1/2 =(v n − √ au n )j+1 − 1/2hδ −n j+1 , δ ± j = 1 h (... |

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Citation Context ... entropy solution. 955 c○1999 American Mathematical Societys956 A. CHALABI Error bounds related to the approximation of (1.1)–(1.2) were derived in [5], [25] and [29]. In a recent study, Jin and Xin (=-=[16]-=-) introduced explicit relaxing schemes for the approximation of systems of hyperbolic conservation laws without source terms. Our purpose in this paper is the study of the convergence of semi-implicit... |

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Citation Context ... √ au n )j+1 − 1/2hδ −n j+1 , δ ± j = 1 h (vj+1 ± √ auj+1 − vj ± √ auj)φ(θ ± j ) θ ± j = vj ± √ auj − vj−1 ± √ auj−1 vj+1 ± √ auj+1 − vj ± √ , auj where φ is a limiter function as defined by Sweby in =-=[27]-=- and satisfies 0 ≤ φ(θ) ≤ 2, θ 0 ≤ φ(θ) ≤ 2. (5.10) Then and u n j+1/2 v n j+1/2 1 = 2 (unj + u n j+1) − 1 2 √ a (vn j+1 − v n j )+ h 4 √ a (δ+n j +δ−n j+1 ), 1 = 2 (vn j + v n √ a j+1) − 2 (unj+1 − u... |

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Citation Context ...HEMES 957 Let u(x, t) =S(t)u0denote the unique weak solution of (1.1)–(1.2) which satisfies the entropy condition. Let us assume that sup q u ′ (2.3) (u) ≤ γ, γ = constant. Using a result of Kruzkov (=-=[18]-=-), we can easily prove the following Proposition 2.1. If u0 ∈ BV (R) ∩ L 1 (R),f ∈C 1 (R),q ∈C 1 (R)such that q(0) = 0 and q ′ ≤ 0, then the problem (1.1)–(1.2) possesses a unique entropy solution u(x... |

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Citation Context ...n in scalar case as a Cauchy problem of the form (1.1) ut + f(u)x = q(u), (x, t) ∈ R×]0,T[; T>0 (1.2) u(x, 0) = u0(x), x ∈ R. Theoretical study of the relaxation phenomena may be found in Chen et al. =-=[7]-=-, Liu [20], and Whitham [32]. Numerical methods have been derived for the approximation of the conservation laws including nonstiff source terms in [3], [4], [24], [26], and [29]. These methods are ba... |

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Citation Context ...nt of the stiff source terms; this motivates the use of semi-implicit and fully implicit schemes. The approximation of the stiff case was recently studied by several authors (see [2], [6], [8], [13], =-=[15]-=-, [17], [19], [25], and [26]), where different methods like asymptotic or splitting methods are used. The main difficulty when we deal with the numerical solution of the stiff problems is the wrong lo... |

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Citation Context ...relaxation phenomena may be found in Chen et al. [7], Liu [20], and Whitham [32]. Numerical methods have been derived for the approximation of the conservation laws including nonstiff source terms in =-=[3]-=-, [4], [24], [26], and [29]. These methods are based on explicit difference schemes. It is well known that explicit schemes are not appropriate for the numerical treatment of the stiff source terms; t... |

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Citation Context ...reatment of the stiff source terms; this motivates the use of semi-implicit and fully implicit schemes. The approximation of the stiff case was recently studied by several authors (see [2], [6], [8], =-=[13]-=-, [15], [17], [19], [25], and [26]), where different methods like asymptotic or splitting methods are used. The main difficulty when we deal with the numerical solution of the stiff problems is the wr... |

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Citation Context ... we suppose that q ′ ≤ 0. This last hypothesis is realistic since it does indicate the dissipativity of the source term q in the sense of [7]. This is the case in the models of combustion ([2], [11], =-=[21]-=-), gas dynamics with heat transfer ([13]), water waves in the presence of the friction force of the bottom ([7]), etc. We point out that in all these examples and many others the source term is not a ... |

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Citation Context ...der the following implicit relaxed scheme: � n vj = f(un j ) u n+1 j = un r j − 2 (f(un+1 j+1 ) − f(un+1 j−1 )) + √ ar 2 (un+1 j+1 − 2un+1 j + un+1 j−1 )+kq(un+1 (5.4) j ). First, we introduce, as in =-=[31]-=-, the operator Tµ defined on piecewise constant functions by Tµ(uh)j = uj − µr 2 (f(uj+1) (5.5) √ aµr − f(uj−1)) + (uj+1 − 2uj + uj−1), j ∈ Z, 2 where µ is a parameter satisfying µ>0, and uh =(uj)j∈Z.... |

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Citation Context ...elaxed scheme (6.5) converges toward the entropy solution of (6.1)–(6.2). Remark 6.1. Second order accurate schemes may be constructed as in subsection 5.3; however these schemes will not be TVD (see =-=[12]-=-). 7. Conclusion We presented an analysis of a class of relaxation schemes for hyperbolic conservation laws including stiff source terms. This method was introduced for the first time by Jin and Xin i... |

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Citation Context ...+ ɛ((a − f ′ (u) 2 )ux)x. Thus, (3.3) is dissipative if the following subcharactrestic condition is satisfied: (3.4) − √ a ≤ f ′ (u) ≤ √ a for all u. Let Q = R×]0,T[, then by similar proof to that in =-=[9]-=-, we prove Theorem 3.1. Suppose that the initial data is bounded and the subcharacteristic condition (3.4) is satisfied, then i) The relaxation problem (3.1)–(3.2) admits a unique solution. ii) If we ... |

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Citation Context ...umerical treatment of the stiff source terms; this motivates the use of semi-implicit and fully implicit schemes. The approximation of the stiff case was recently studied by several authors (see [2], =-=[6]-=-, [8], [13], [15], [17], [19], [25], and [26]), where different methods like asymptotic or splitting methods are used. The main difficulty when we deal with the numerical solution of the stiff problem... |

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Citation Context ...e found in Chen et al. [7], Liu [20], and Whitham [32]. Numerical methods have been derived for the approximation of the conservation laws including nonstiff source terms in [3], [4], [24], [26], and =-=[29]-=-. These methods are based on explicit difference schemes. It is well known that explicit schemes are not appropriate for the numerical treatment of the stiff source terms; this motivates the use of se... |

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Citation Context ...ation phenomena may be found in Chen et al. [7], Liu [20], and Whitham [32]. Numerical methods have been derived for the approximation of the conservation laws including nonstiff source terms in [3], =-=[4]-=-, [24], [26], and [29]. These methods are based on explicit difference schemes. It is well known that explicit schemes are not appropriate for the numerical treatment of the stiff source terms; this m... |

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Citation Context ...the stiff source terms; this motivates the use of semi-implicit and fully implicit schemes. The approximation of the stiff case was recently studied by several authors (see [2], [6], [8], [13], [15], =-=[17]-=-, [19], [25], and [26]), where different methods like asymptotic or splitting methods are used. The main difficulty when we deal with the numerical solution of the stiff problems is the wrong location... |

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Citation Context ...the numerical treatment of the stiff source terms; this motivates the use of semi-implicit and fully implicit schemes. The approximation of the stiff case was recently studied by several authors (see =-=[2]-=-, [6], [8], [13], [15], [17], [19], [25], and [26]), where different methods like asymptotic or splitting methods are used. The main difficulty when we deal with the numerical solution of the stiff pr... |

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route de Narbonne 31062 Toulouse cedex France E-mail address: chalabi@mip.ups-tlse.fr
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Citation Context ...y problem of the form (1.1) ut + f(u)x = q(u), (x, t) ∈ R×]0,T[; T>0 (1.2) u(x, 0) = u0(x), x ∈ R. Theoretical study of the relaxation phenomena may be found in Chen et al. [7], Liu [20], and Whitham =-=[32]-=-. Numerical methods have been derived for the approximation of the conservation laws including nonstiff source terms in [3], [4], [24], [26], and [29]. These methods are based on explicit difference s... |

1 |
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Citation Context ...eme, semi-implicit scheme, TVD scheme, MUSCL method, entropy solution. 955 c○1999 American Mathematical Societys956 A. CHALABI Error bounds related to the approximation of (1.1)–(1.2) were derived in =-=[5]-=-, [25] and [29]. In a recent study, Jin and Xin ([16]) introduced explicit relaxing schemes for the approximation of systems of hyperbolic conservation laws without source terms. Our purpose in this p... |