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

Venue: | Math. Comput |

Citations: | 8 - 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.

### Citations

576 |
Towards the ultimate conservative difference scheme, V. A second order sequel to Godunov’s method
- Leer
- 1979
(Show Context)
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 (... |

198 |
High resolution schemes using flux limiters for hyperbolic conservation laws
- Sweby
- 1984
(Show Context)
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... |

197 | The Relaxation Scheme for Systems of Conservation Laws in Arbitrary Space Dimension
- Jin, Xin
- 1995
(Show Context)
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... |

156 |
First order quasi-linear equations in several independent variables
- Kružkov
- 1970
(Show Context)
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... |

143 | Hyperbolic conservation laws with stiff relaxation terms and entropy
- Chen, Levermore, et al.
- 1994
(Show Context)
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... |

136 |
Hyperbolic conservation laws with relaxation
- Liu
- 1987
(Show Context)
Citation Context ...ar 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 based on exp... |

93 | Monotone difference approximations for scalar conversation laws - Grandall, Majda - 1980 |

91 | Riemann solvers, the entropy condition, and difference approximations - Osher - 1984 |

78 | Convergence to equilibrium for the relaxation approximation of conservation laws - Natalini - 1996 |

62 | Numerical schemes for hyperbolic conservation laws with stiff relaxation terms
- 25Jin, Levermore
- 1996
(Show Context)
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... |

61 | Numerical Methods for Hyperbolic Conservation laws with stiff relaxation
- Pember
(Show Context)
Citation Context ... 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 motivat... |

59 | Numerical viscosity and the entropy condition for conservative difference schemes - Tadmor - 1984 |

58 |
Upwind methods for hyperbolic conservation laws with source terms. Computers Fluids
- Bermudez, Vazquez
- 1994
(Show Context)
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... |

57 |
A study of numerical methods for hyperbolic conservation laws with sti source terms
- LeVeque, Yee
- 1990
(Show Context)
Citation Context ...iff 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 of th... |

50 | Runge-Kutta methods for hyperbolic conservation laws with stiff relaxation terms
- Jin
- 1995
(Show Context)
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... |

47 |
Theoretical and numerical structure for reacting shock waves
- Colella, Majda, et al.
- 1986
(Show Context)
Citation Context ...cal 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 problems is ... |

36 | Convergence of relaxation schemes for conservation laws - Aregba-Driollet, Natalini - 1996 |

35 |
A qualitative model for dynamic combustion
- Majda
- 1981
(Show Context)
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 ... |

34 |
Convergence and error estimates in finite volume schemes for general multidimensional scalar conservation laws. I. Explicit monotone schemes, Mathematical Modelling and Numerical Analysis 28
- Vila
- 1994
(Show Context)
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.... |

25 |
Robust Difference approximation of stiff inviscid detonation waves, manuscript
- Engquist, Sjogreen
(Show Context)
Citation Context ...ting methods are used. The main difficulty when we deal with the numerical solution of the stiff problems is the wrong location of the discontinuities. This problem has been investigated in [2], [8], =-=[11]-=-, [13], [15], [17], and [19]. Received by the editor April 29, 1997 and, in revised form, October 14, 1997. 1991 Mathematics Subject Classification. Primary 35L65, 65M05, 65M10. Key words and phrases.... |

25 |
On the accuracy of stable schemes for 2D scalar conservation laws
- Goodman, LeVeque
- 1985
(Show Context)
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... |

13 | On convergence of numerical schemes for hyperbolic conservation laws with stiff source terms
- Chalabi
- 1997
(Show Context)
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... |

13 |
Convergence of the relaxation approximation to a scalar nonlinear hyperbolic equation arising in chromatography
- Collet, Rascle
- 1996
(Show Context)
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 ... |

13 | Error bounds for fractional step methods for conservation laws with source terms
- Tang, Teng
- 1995
(Show Context)
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... |

10 |
An L 1 error bound for a semi-implicit difference scheme applied to a stiff system of conservation laws
- Schroll, Tveito, et al.
- 1997
(Show Context)
Citation Context ...urce 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 of the disc... |

6 |
Stable upwind schemes for hyperbolic conservation laws with source, terms
- Chalabi
- 1992
(Show Context)
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... |

6 | A convex entropy for a hyperbolic system with stiff relaxation
- Jin
- 1996
(Show Context)
Citation Context ...ɛ ) − v ɛ || L 2 (Q)≤ C √ ɛ, Proof. The proof of i) is based on invariant regions (see [9]). Using the entropy inequality we obtain ii). Let us prove property iii) as we do in the homogeneous case in =-=[14]-=-. Let (η(u, v),F(u, v)) be a convex entropy-flux entropy function associated with the system (3.1) such that aηv = Fu, ηu = Fv by multiplying the first equation of (3.1) by ηu and the second equation ... |

6 |
Hyperbolic Conservation Laws with Source Terms: Errors of the Shock Location, Research report 94-07
- Klingenstein
- 1994
(Show Context)
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... |

3 |
Thije Boonkkamp, The numerical wave speed for one-dimensional scalar hyperbolic conservation laws with source terms, RANA 94-01, Eindhover Univ. of Tech
- Berkenbosch, Kaasschieter, et al.
- 1994
(Show Context)
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... |

2 |
Finite difference schemes for conservation laws with source terms
- Schroll, Winther
- 1996
(Show Context)
Citation Context ...mena 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 motivates the... |

2 |
route de Narbonne 31062 Toulouse cedex France E-mail address: chalabi@mip.ups-tlse.fr
- Whitham, Linear, et al.
- 1974
(Show Context)
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 |
An error bound for the polygonal approximation of conservation laws with source terms
- Chalabi
- 1996
(Show Context)
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... |