## Localization effects and measure source terms in numerical schemes for balance laws

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Venue: | Math. Comp |

Citations: | 13 - 3 self |

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@ARTICLE{Gosse_localizationeffects,

author = {Laurent Gosse},

title = {Localization effects and measure source terms in numerical schemes for balance laws},

journal = {Math. Comp},

year = {},

pages = {2002}

}

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### Abstract

Abstract. This paper investigates the behavior of numerical schemes for nonlinear conservation laws with source terms. We concentrate on two significant examples: relaxation approximations and genuinely nonhomogeneous scalar laws. The main tool in our analysis is the extensive use of weak limits and nonconservative products which allow us to describe accurately the operations achieved in practice when using Riemann-based numerical schemes. Some illustrative and relevant computational results are provided. 1.

### Citations

164 | The relaxation schemes for systems of conservation laws in arbitrary space dimensions
- Jin, Xin
- 1995
(Show Context)
Citation Context ...perbolic equations written in conservation form, [31]. We will focus on two significant examples. The first one is the special case of relaxation approximation to the scalar conservation law (1)–(2), =-=[28, 29]-=-. The second one corresponds to the general case of a scalar balance law (26). These two problems share the common feature of being solvable within the framework of BV functions and Kruˇzkov’s theory,... |

98 |
The spaces BV and quasilinear equations
- Vol’pert
(Show Context)
Citation Context .... The second one corresponds to the general case of a scalar balance law (26). These two problems share the common feature of being solvable within the framework of BV functions and Kruˇzkov’s theory,=-=[31,41]-=-. In both cases, when designing any numerical scheme, one has to face the issue of choosing an efficient treatment of the zero-order term. The constraints are mostly stability in stiff regimes (as poi... |

77 |
Monotone Difference Approximations for Scalar Conservation Laws
- Crandall, Majda
- 1980
(Show Context)
Citation Context ...ns prescribed in Theorem 1. Therefore, we can pass to the limits µ∆t → +∞, h → 0. The entropy condition is enforced by the monotonicity property of this relaxed scheme and we can apply the results of =-=[10]-=-. 2.6. Numerical results. We performed some computational experiments on the following test case with these two initial data: � � 2 u ∂tu + ∂x =0withx∈R and 0 <t≤ 0.3, (25) u0(x) = 2 � 1 if x<0, 0 if ... |

74 |
Definition and weak stability of nonconservative products
- Maso, LeFloch, et al.
- 1995
(Show Context)
Citation Context ... ambiguous nonconservative products. Since weak solutions of hyperbolic equations are likely to be discontinuous, their product with Dirac measures is unstable and the relevant theory is developed in =-=[5, 9, 12, 24, 32, 36]-=-. At the numerical level, the key point is to be able to compute accurately such measure-valued terms: some attempts have already been proposed in [17, 19, 20, 22, 23]; see also [8, 21, 40, 39] for cl... |

66 |
A well balanced scheme for the numerical processing of source terms in hyperbolic equations
- Greenberg, Roux
- 1996
(Show Context)
Citation Context ... computational viewpoint is the Riemann problem for (49) corresponding to initial data of the form: � uL if x<0, u(x, 0) = uR if x>0. Its entropy solution is made of two elementary waves; we refer to =-=[19, 23]-=- for more details in this direction. The general results of [12, 32, 36] guarantee uniqueness for these Riemann problems, but the next lemma establishes uniqueness for the Cauchy problem (49) under so... |

64 |
First order quasi-linear equations in several independent variables
- Kruˇzkov
- 1970
(Show Context)
Citation Context ...ve and relevant computational results are provided. 1. Introduction This paper deals with the numerical analysis of weak solutions to nonhomogeneous hyperbolic equations written in conservation form, =-=[31]-=-. We will focus on two significant examples. The first one is the special case of relaxation approximation to the scalar conservation law (1)–(2), [28, 29]. The second one corresponds to the general c... |

50 | One-dimensional transport equations with discontinuous coecients
- Bouchut, James
- 1998
(Show Context)
Citation Context ... ambiguous nonconservative products. Since weak solutions of hyperbolic equations are likely to be discontinuous, their product with Dirac measures is unstable and the relevant theory is developed in =-=[5, 9, 12, 24, 32, 36]-=-. At the numerical level, the key point is to be able to compute accurately such measure-valued terms: some attempts have already been proposed in [17, 19, 20, 22, 23]; see also [8, 21, 40, 39] for cl... |

44 |
Generalized characteristics and the structure of solutions of hyperbolic conservation laws
- Dafermos
- 1977
(Show Context)
Citation Context ...towards the entropy solution of (26) in the context of BV functions. 3.2. Some results about initial-boundary-value problems. Before entering the core of the matter, we recall some basic results from =-=[3, 11, 35]-=- concerning the boundary value problem for (26): ⎧ ⎨ ∂tu + ∂xf(u) =k(x)g(u) withx∈ ] − L, L[, t>0, u(., 0) = u0 ∈ BV (−L, L), ⎩ u(−L, t) =uL(t), u(L, t) =uR(t) inBV (R + (27) ).sLOCALIZATION EFFECTS A... |

44 |
Convex conservation laws with discontinuous coefficients: Existence, uniqueness and asymptotic behavior
- Klingenberg, Risebro
- 1995
(Show Context)
Citation Context ...e, it is always possible to reverse the signs. The point here is to avoid any resonant situation f ′ (u ε ) = 0: we refer to [27, 34] for a study of resonance in the context of balance laws; see also =-=[13, 30]-=-. 1 Suppose u → ū in R, then|φ ′ (u)| →+∞. Since a ∈ R, φ −1 ◦ (φ(u) − a) → φ −1 ◦ φ(ū) =ū. Conversely, if w(ū, a) =ū, we apply φ to both sides: if a �= 0, this means that |φ(ū)| diverges and g(ū) = 0... |

39 | Tzavaras, Contractive relaxation systems and the scalar multidimensional conservation law
- Katsoulakis, E
- 1997
(Show Context)
Citation Context ...perbolic equations written in conservation form, [31]. We will focus on two significant examples. The first one is the special case of relaxation approximation to the scalar conservation law (1)–(2), =-=[28, 29]-=-. The second one corresponds to the general case of a scalar balance law (26). These two problems share the common feature of being solvable within the framework of BV functions and Kruˇzkov’s theory,... |

35 |
Why nonconservative schemes converge to wrong solutions: Error analysis
- Hou, Floch
- 1994
(Show Context)
Citation Context ...ccurately such measure-valued terms: some attempts have already been proposed in [17, 19, 20, 22, 23]; see also [8, 21, 40, 39] for closely related works. We stress that our results do not contradict =-=[26]-=-. The paper is organized as follows. In Section 2, we consider the relaxing approximations with a time-based localization function. We show compactness properties as this function concentrates. A Godu... |

34 |
Systems of conservation laws with invariant submanifolds
- Temple
- 1983
(Show Context)
Citation Context ...) f ′ (u) � , � � 1 Rf ′ = 0 Thus, one can check that w(u, a), (37), is a Riemann invariant and that (47) belongs to the Temple class (in the sense that wave curves are Riemann invariant level sets), =-=[38]-=-, once (41), (42), (43) rule the discontinuity points of a through some generalized jump relations. Our BV-bound on w(u ε ,a ε ) (see the proof of Lemma 7) was somehow predictable from this perspectiv... |

33 | Convergence of relaxation schemes for conservation laws
- Aregba-Driollet, Natalini
- 1996
(Show Context)
Citation Context ...thms. 2.2. Introduction of a singular relaxation system. Let us begin with a remark which will be of constant use in the sequel: thanks to its linear convective part, the system (3) partly decouples, =-=[1]-=-: � ∂twε + c∂xwε = µG(wε ,zε )∂tbε (t), ∂tzε − c∂xzε = −µG(wε ,zε )∂tbε (4) (t), where: w = −(v + cu); z = v − cu; G(w, z) = 1 � � w + z (5) (−w + z) − f − . 2 2c Without any loss of generality, we ca... |

33 | Construction of BGK models with a family of kinetic entropies for a given system of conservation laws
- Bouchut
- 1999
(Show Context)
Citation Context ...,zn j ) � G(w n j−1 ,zn j ) − 2G(wn j ,zn j )+G(wn j ,zn j+1 ) − c∆t (1 − exp(−µ∆t)) . h At this level, we point out that this approach is quite close to the one which has been studied by F. Bouchut, =-=[4]-=- and A. Vasseur, [40], in the context of kinetic equations. It is also related to the modified split-scheme proposed in [25] in the context of reactive Euler equations. At last, the split-schemes inve... |

29 |
On the Relation Between the Upwind-Differencing Schemes of Godunov, Enguist-Osher and Roe
- Leer
- 1985
(Show Context)
Citation Context ...[5, 9, 12, 24, 32, 36]. At the numerical level, the key point is to be able to compute accurately such measure-valued terms: some attempts have already been proposed in [17, 19, 20, 22, 23]; see also =-=[8, 21, 40, 39]-=- for closely related works. We stress that our results do not contradict [26]. The paper is organized as follows. In Section 2, we consider the relaxing approximations with a time-based localization f... |

22 |
Convergence of the 2 × 2 Godunov method for a general resonant nonlinear balance law
- Isaacson, Temple
- 1995
(Show Context)
Citation Context ...r program is carried out for a general scalar balance law with a localization function depending on the space variable. In this last case, one has to face new issues as nonlinear resonance may occur, =-=[27]-=-, which we shall systematically exclude with convenient hypotheses. In Section 4, we conclude and propose some perspectives for future developments. 2. The special case of relaxation approximations 2.... |

21 | A well-balanced scheme using non-conservative products designed for hyperbolic systems of conservation laws with source terms
- Gosse
(Show Context)
Citation Context ...elevant theory is developed in [5, 9, 12, 24, 32, 36]. At the numerical level, the key point is to be able to compute accurately such measure-valued terms: some attempts have already been proposed in =-=[17, 19, 20, 22, 23]-=-; see also [8, 21, 40, 39] for closely related works. We stress that our results do not contradict [26]. The paper is organized as follows. In Section 2, we consider the relaxing approximations with a... |

19 | Well-posedness for a class of 2 × 2 conservation laws with L ∞ data
- Baiti, Jenssen
- 1997
(Show Context)
Citation Context ... BV-bound on w(u ε ,a ε ) (see the proof of Lemma 7) was somehow predictable from this perspective. A “conservative version” of (47), � ∂tu + ∂xf(u, a) =0, (48) ∂ta =0 has been studied extensively in =-=[2]-=- in strictly hyperbolic case, i.e., under the assumption that ∂uf(u, a) > 0 in the context of L 1 ∩ L ∞ functions. The general situation including resonant regimes has been tackled by the authors of [... |

19 |
Nonlinear resonance for quasilinear hyperbolic equation
- Liu
- 1987
(Show Context)
Citation Context ...eneral. At this point, we have a problem since f ′ (0) = g(0) = 0. Of course, it is always possible to reverse the signs. The point here is to avoid any resonant situation f ′ (u ε ) = 0: we refer to =-=[27, 34]-=- for a study of resonance in the context of balance laws; see also [13, 30]. 1 Suppose u → ū in R, then|φ ′ (u)| →+∞. Since a ∈ R, φ −1 ◦ (φ(u) − a) → φ −1 ◦ φ(ū) =ū. Conversely, if w(ū, a) =ū, we app... |

18 |
The Convergence Rate of finite Difference Schemes in the Presence of Shocks
- Engquist, Sjogreen
(Show Context)
Citation Context ...see, e.g., [18, 33] and references therein; • the solutions one gets at numerical steady-state may be very poor approximations of the expected large time behavior of the original equation: see, e.g., =-=[7,8,13,14,19,20]-=-. Recently, Greenberg and LeRoux have proposed to treat the particular case g(u) =−u in a different way inside a Godunov type scheme, [23]. One motivation was to get rid of the error coming from the p... |

15 |
A modified fractional step method for the accurate approximation of detonation waves
- Helzel, LeVeque, et al.
- 2000
(Show Context)
Citation Context ...his approach is quite close to the one which has been studied by F. Bouchut, [4] and A. Vasseur, [40], in the context of kinetic equations. It is also related to the modified split-scheme proposed in =-=[25]-=- in the context of reactive Euler equations. At last, the split-schemes investigated in [1] are obtained just by inserting a projection stage onto piecewise constant functions similar to (19) between ... |

12 | Representation of weak limits and definition of non-conservative products - LeFloch, Tzavaras - 1999 |

11 | Error bounds for fractional step methods for conservation laws with source terms
- Tang, Teng
- 1995
(Show Context)
Citation Context ... The advantages of this nonconservative approach are clearly noticeable even if theoretical results guarantee the convergence of the classical time-splitting scheme as the grid is refined, see, e.g., =-=[37]-=-. We refer to [8, 19, 23] for other comparisons between well-balanced approaches and more conventional discretizations on more complex and realistic problems. 4. Conclusion We proposed in this paper a... |

10 | A priori error estimate for a well-balanced scheme designed for inhomogeneous scalar conservation laws
- Gosse
- 1998
(Show Context)
Citation Context ...elevant theory is developed in [5, 9, 12, 24, 32, 36]. At the numerical level, the key point is to be able to compute accurately such measure-valued terms: some attempts have already been proposed in =-=[17, 19, 20, 22, 23]-=-; see also [8, 21, 40, 39] for closely related works. We stress that our results do not contradict [26]. The paper is organized as follows. In Section 2, we consider the relaxing approximations with a... |

10 |
Kinetic semi-discretization of scalar conservation laws and convergence by using averaging lemmas
- Vasseur
- 1999
(Show Context)
Citation Context ...[5, 9, 12, 24, 32, 36]. At the numerical level, the key point is to be able to compute accurately such measure-valued terms: some attempts have already been proposed in [17, 19, 20, 22, 23]; see also =-=[8, 21, 40, 39]-=- for closely related works. We stress that our results do not contradict [26]. The paper is organized as follows. In Section 2, we consider the relaxing approximations with a time-based localization f... |

9 |
Multiple steady state for 1−D transonic flow
- Embid, Goodman, et al.
- 1984
(Show Context)
Citation Context ... regime should occur in order to respect the dynamics of the differential underlying problem, [35]. It turns out that these two criteria are actually difficult to meet within the existing approaches, =-=[7, 8, 13, 23]-=-. One goal of the present article is to propose to work out both objectives efficiently in a simple self-contained Godunov approach. The basic trick is therefore to consider special Riemann problems e... |

9 |
Un schéma-équilibre adapté aux lois de conservation scalaires non-homogènes
- Gosse, LeRoux
- 1996
(Show Context)
Citation Context ...elevant theory is developed in [5, 9, 12, 24, 32, 36]. At the numerical level, the key point is to be able to compute accurately such measure-valued terms: some attempts have already been proposed in =-=[17, 19, 20, 22, 23]-=-; see also [8, 21, 40, 39] for closely related works. We stress that our results do not contradict [26]. The paper is organized as follows. In Section 2, we consider the relaxing approximations with a... |

7 |
Discontinous generalized solutions of nonlinear nonconservative hyperbolic equation
- Cauret, Colombeau, et al.
- 1989
(Show Context)
Citation Context ... ambiguous nonconservative products. Since weak solutions of hyperbolic equations are likely to be discontinuous, their product with Dirac measures is unstable and the relevant theory is developed in =-=[5, 9, 12, 24, 32, 36]-=-. At the numerical level, the key point is to be able to compute accurately such measure-valued terms: some attempts have already been proposed in [17, 19, 20, 22, 23]; see also [8, 21, 40, 39] for cl... |

5 |
A.-Y.: Un schéma équilibre adapté au modèle d’atmosphère avec termes de gravité
- Cargo, Roux
- 1994
(Show Context)
Citation Context ... regime should occur in order to respect the dynamics of the differential underlying problem, [35]. It turns out that these two criteria are actually difficult to meet within the existing approaches, =-=[7, 8, 13, 23]-=-. One goal of the present article is to propose to work out both objectives efficiently in a simple self-contained Godunov approach. The basic trick is therefore to consider special Riemann problems e... |

5 |
A well-balanced flux splitting scheme designed for hyperbolic systems of conservation laws with source terms
- Gosse
- 2000
(Show Context)
Citation Context ...that it is not defined aprioriand for any arbitrary choice of u and a h . This is somehow a uniqueness result concerning the choice of the family of paths used in the numerical algorithms proposed in =-=[19, 20, 22]-=-.s572 LAURENT GOSSE An alternative formulation of (34) also used in, e.g., [27, 19] is as follows: � ∂tu + ∂xf(u) − g(u)∂xa =0, (47) ∂ta =0. This “system” is not ambiguous once the initial datum for a... |

5 |
Nonconservative products in bounded variation functions
- Colombeau, Heibig
- 1992
(Show Context)
Citation Context |

4 |
Recent developments of the GRP method
- Falcovitz, Ben-Artzi
- 1995
(Show Context)
Citation Context ...listic any longer since it leads to intricate and computationally expensive schemes based mostly on generalized Riemann problems whose structure is highly more intricate compared to homogeneous ones, =-=[15, 16]-=-. One is therefore led to tackle some singular localization functions, describing the fact that the relaxing or reactive processes are ignited and switched off suddently at particular moments or locat... |

4 |
A generalized Riemann problem for quasi one dimensional gas flows
- Glimm, Marshall, et al.
- 1984
(Show Context)
Citation Context ...listic any longer since it leads to intricate and computationally expensive schemes based mostly on generalized Riemann problems whose structure is highly more intricate compared to homogeneous ones, =-=[15, 16]-=-. One is therefore led to tackle some singular localization functions, describing the fact that the relaxing or reactive processes are ignited and switched off suddently at particular moments or locat... |

3 |
Numerical approximation of linear one-dimensional conservation equations with discontinuous coefficients
- Gosse, James
(Show Context)
Citation Context ...[5, 9, 12, 24, 32, 36]. At the numerical level, the key point is to be able to compute accurately such measure-valued terms: some attempts have already been proposed in [17, 19, 20, 22, 23]; see also =-=[8, 21, 40, 39]-=- for closely related works. We stress that our results do not contradict [26]. The paper is organized as follows. In Section 2, we consider the relaxing approximations with a time-based localization f... |

3 |
Long-time behavior for conservation laws with source in a bounded domain
- Mascia, Terracina
- 1999
(Show Context)
Citation Context ...(the local Maxwellian) is the key point and the general balance law, where a longtime decay to some steady regime should occur in order to respect the dynamics of the differential underlying problem, =-=[35]-=-. It turns out that these two criteria are actually difficult to meet within the existing approaches, [7, 8, 13, 23]. One goal of the present article is to propose to work out both objectives efficien... |

2 |
First-order quasilinear equations with boundary conditions
- Bardos, Leroux, et al.
- 1979
(Show Context)
Citation Context ...towards the entropy solution of (26) in the context of BV functions. 3.2. Some results about initial-boundary-value problems. Before entering the core of the matter, we recall some basic results from =-=[3, 11, 35]-=- concerning the boundary value problem for (26): ⎧ ⎨ ∂tu + ∂xf(u) =k(x)g(u) withx∈ ] − L, L[, t>0, u(., 0) = u0 ∈ BV (−L, L), ⎩ u(−L, t) =uL(t), u(L, t) =uR(t) inBV (R + (27) ).sLOCALIZATION EFFECTS A... |

2 |
Kruˇzkov’s inequalities for scalar conservation laws revisited
- Bouchut, Perthame
- 1998
(Show Context)
Citation Context ...0. The entropy inequality becomes: � � � |u(x, t) − v(x, t)|∂tψ(x, t)+sgn(u−v)(f(u) − f(v))∂xψ(x, t) .dx.dt ≥ 0. R×R + ∗ Now, it remains to select ψ using regularized Heaviside functions as in, e.g., =-=[31,6, 29]-=-; the fluxes and the remaining terms cancel. We finally derive: � |u(x, t) − v(x, t)|∂tψ(x, t).dx.dt ≥ 0. R×R + ∗ We close this section with the construction of S, thesolution operator for (49) mappin... |

2 | la stabilité des approximations implicites des lois de conservation scalaires non-homogènes - Gosse, Sur - 1999 |

2 |
A new definition of nonconservative products and weak stability results
- Raymond
- 1996
(Show Context)
Citation Context ... GOSSE 3.4. The quest for the weak-⋆ limit. Now we want to shed some light on the ambiguous term emanating from the limit ε → 0 of (34). Once again, we will use nonconservative products as defined in =-=[32, 36]-=-, that is, as the weak-⋆ limits of the compact sequences g(u ε )∂xa ε . The next result is therefore a corner stone of our construction. Proposition 5. Under the assumptions of Lemma 7, there holds: (... |

1 |
Convergence of flux-limiting schemes for hyperbolic conservation laws with source terms
- Burton
- 1993
(Show Context)
Citation Context ... regime should occur in order to respect the dynamics of the differential underlying problem, [35]. It turns out that these two criteria are actually difficult to meet within the existing approaches, =-=[7, 8, 13, 23]-=-. One goal of the present article is to propose to work out both objectives efficiently in a simple self-contained Godunov approach. The basic trick is therefore to consider special Riemann problems e... |

1 |
A study of numerical methods for hyperbolic equations with stiff source terms
- LeVeque, Yee
- 1990
(Show Context)
Citation Context ...en designing any numerical scheme, one has to face the issue of choosing an efficient treatment of the zero-order term. The constraints are mostly stability in stiff regimes (as pointed out in, e.g., =-=[18, 33]-=-) together with some consistency with an expected asymptotic behavior. For our examples, we distinguished between relaxation, for which the convergence towards the equilibrium manifold (the local Maxw... |