On convergence of numerical schemes for hyperbolic conservation laws with stiff source terms

Chalabi, A.

doi:10.1090/s0025-5718-97-00817-x

Cited by 26 publications

(38 citation statements)

References 23 publications

(28 reference statements)

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“…By a similar proof as in [6], we make use of Helly's theorem to get a subsequence extracted from (u h ) which converges toward the entropy solution of (1.1)-(1.2), and the uniqueness of the entropy satisfying solution ensures that all the sequence (u h ) converges toward the entropy satisfying solution u of problem (1.1)-(1.2) as h tends toward zero.…”

Section: Convergence Of Relaxation Schemes 961mentioning

confidence: 99%

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Convergence of relaxation schemes for hyperbolic conservation laws with stiff source terms

Chalabi

1999

Math. Comp.

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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.

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Section: Convergence Of Relaxation Schemes 961mentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

Convergence of relaxation schemes for hyperbolic conservation laws with stiff source terms

Chalabi

1999

Math. Comp.

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“…Approximate numerical solutions given by scheme (7) to the Linear hyperbolic PDE (6), are shown in Figure 1, along with smooth Gaussian initial condition ( , 0) = ( ) = − 2 . It is shown initial condition (top left) and computed solutions at t = 2 (top right), t = 4 (bottom left) and t = 8 (bottom right) with CFL number = where the diffusion is in balance or dominates the dispersion, the above numerical experiments related to those in Figure 1, illustrate the fact of entire truncation error vanishing (see right picture) at all grid points under grid refinement, as expected from previous theoretical analysis.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…For example, when the source term is a non-increasing function, the total variation of the exact solution of the scalar balance law is also a nonincreasing function, as in the homogeneous case (see, e.g., [16,10]). In general, however, the source term might not be decreasing (see [6,11,27]) and some semiimplicit and fully implicit scheme are not applicable, at least in a straightforward manner [6,11].…”

Section: Introductionmentioning

confidence: 99%

Lagrangian-Eulerian approximation methods for balance laws and hyperbolic conservation laws

Abreu

Pérez²,

Santo³

2018

revuin

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Este artículo puede compartirse bajo la licencia CC BY-ND 4.0 y se referencia usando el siguiente formato: E. Abreu, J. Pérez, A. Santo, "Lagrangian-Eulerian approximation methods for balance laws and hyperbolic conservation laws," Rev. UIS Ing., vol. 17, no. 1, pp. 191-200, 2018. Doi: https://doi.org/10.18273/revuin.v17n1-2018018 http://revistas.uis.edu.co/index.php/revistauisingenierias Vol. 17, no. 1, pp. 191-200, enero-junio, 2018 Revista UIS Ingenierías RESUMENUn nuevo volumen finito de control es presentado en un enfoque Lagrangiano-Euleriano (ver artículos [1,28]), en este, un dominio de espacio-tiempo es estudiado con el fin de diseñar un esquema localmente conservativo. Tal esquema tiene en cuenta el delicado balance no linear, entre las aproximaciones numéricas del flujo hiperbólico y el término fuente, en problemas de ley de balance ligados con leyes de conservación puramente hiperbólicas. Además, combinando algunas ideas de este nuevo enfoque, hacemos una construcción formal de un nuevo algoritmo para resolver importantes problemas de leyes de conservación en dos dimensiones espaciales. Un conjunto pertinente de experimentos numéricos para diferentes modelos es presentado para mostrar evidencia que soluciones cualitativamente correctas son aproximadas. PALABRAS CLAVE:Leyes de conservación; Lagrangiano-Euleriano; volumen finito. ABSTRACTA new finite control volume in a Lagrangian-Eulerian framework is presented (see papers [1,28]), in which a local space-time domain is studied, in order to design a locally conservative scheme. Such scheme accounts for the delicate nonlinear balance between the numerical approximations of the hyperbolic flux and the source term for balance law problems linked to the purely hyperbolic character of conservation laws. Furthermore, by combining the ideas of this new approach, we give a formal construction of a new algorithm for solving several nonlinear hyperbolic conservation laws in two space dimensions. Here, a set of pertinent numerical experiments for distinct models is presented to evidence that we are calculating the correct qualitatively good solutions.

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“…We point out that, due to the introduction of piecewise linear reconstructions of the function z, the differences of discrete interfacial values approximate the second order derivative, as given in (4.10), and the upwind part (4.23) of the discretization (2.13) "overtakes" the desired result; an additional term (4.27) is thus needed to recover the first order derivatives from the Taylor expansions of the source term (4.22). Moreover, some restrictions (4.31) on the definition of the slope limiter are also required, to guarantee the occurrence of suitable error estimates (refer also to [5], [21] and [22]). Without these assumptions, only suboptimal results are derived (see [18] and [34], for instance).…”

Section: Remarks and Numerical Evidencementioning

confidence: 99%

First and second order error estimates for the Upwind Source at Interface method

Katsaounis

Simeoni

2004

Math. Comp.

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Abstract. The Upwind Source at Interface (U.S.I.) method for hyperbolic conservation laws with source term introduced by Perthame and Simeoni is essentially first order accurate. Under appropriate hypotheses of consistency on the finite volume discretization of the source term, we prove L p -error estimates, 1 ≤ p <+∞, in the case of a uniform spatial mesh, for which an optimal result can be obtained. We thus conclude that the same convergence rates hold as for the corresponding homogeneous problem. To improve the numerical accuracy, we develop two different approaches of dealing with the source term and we discuss the question of deriving second order error estimates. Numerical evidence shows that those techniques produce high resolution schemes compatible with the U.S.I. method.

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On convergence of numerical schemes for hyperbolic conservation laws with stiff source terms

Cited by 26 publications

References 23 publications

Convergence of relaxation schemes for hyperbolic conservation laws with stiff source terms

Convergence of relaxation schemes for hyperbolic conservation laws with stiff source terms

Lagrangian-Eulerian approximation methods for balance laws and hyperbolic conservation laws

First and second order error estimates for the Upwind Source at Interface method

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