Error Estimates for Finite Volume Approximations of Classical Solutions for Nonlinear Systems of Hyperbolic Balance Laws

Jovanović, Vladimir; Rohde, Christian

doi:10.1137/s0036142903438136

Cited by 24 publications

(19 citation statements)

References 28 publications

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“…cf. [10,15,30,32,35]. The missing piece of information between (7.8) and (7.9) is provided by the careful analysis of the renormalized entropy inequality in Section 4, see Lemma 4.3.…”

Section: Energy Balancementioning

confidence: 99%

“…In [35] Jovanović and Rohde assumed the existence of a classical solution to the Cauchy problem of a general multidimensional hyperbolic conservation law. Applying the stability result for classical solutions in the class of entropy solutions due to Dafermos [17] and DiPerna's method [22,23], they derived error estimates for the explicit finite volume schemes satisfying the discrete entropy inequality and thus proved that the numerical solutions convergence strongly to the exact classical solution.…”

Section: Introductionmentioning

confidence: 99%

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Convergence of Finite Volume Schemes for the Euler Equations via Dissipative Measure-Valued Solutions

2019

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The Cauchy problem for the complete Euler system is in general ill posed in the class of admissible (entropy producing) weak solutions. This suggests there might be sequences of approximate solutions that develop fine scale oscillations. Accordingly, the concept of measure-valued solution that capture possible oscillations is more suitable for analysis. We study the convergence of a class of entropy stable finite volume schemes for the barotropic and complete compressible Euler equations in the multidimensional case. We establish suitable stability and consistency estimates and show that the Young measure generated by numerical solutions represents a dissipative measure-valued solution of the Euler system. Here dissipative means that a suitable form of the Second law of thermodynamics is incorporated in the definition of the measure-valued solutions. In particular, using the recently established weak-strong uniqueness principle, we show that the numerical solutions converge pointwise to the regular solution of the limit systems at least on the lifespan of the latter.

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“…cf. [10,15,30,32,35]. The missing piece of information between (7.8) and (7.9) is provided by the careful analysis of the renormalized entropy inequality in Section 4, see Lemma 4.3.…”

Section: Energy Balancementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Convergence of Finite Volume Schemes for the Euler Equations via Dissipative Measure-Valued Solutions

2019

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“…3. An L 2 -contraction estimate for finite volume schemes has been proven in Jovanović and Rohde (2006), and the use of compensated compactness methods (Murat 1981) for numerical approximations is well established. This would be the basis to transfer the analysis of the continuous model to the discrete case.…”

Section: Discretizationmentioning

confidence: 99%

A Hyperbolic–Elliptic Model Problem for Coupled Surface–Subsurface Flow

2015

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We consider a model problem for coupled surface-subsurface flow. The model consists of a nonlinear kinematic wave equation for the surface fluid's height and a Brinkman model that governs fluid velocity and pressure for subsurface dynamics. For this coupled hyperbolic-elliptic model we establish the existence of weak solutions. The proof is based on a viscous approximation and the method of compensated compactness by virtue of appropriate energy estimates. To solve the coupled problem numerically, a finite volume method is applied. The numerical scheme is used to illustrate the influence of the Brinkman parameter on the coupled flow pattern for infiltration scenarios.

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“…Moreover the theoretical study of convergence of these methods for nonlinear transport equations has been addressed in a large amount of papers, most of them based on the Kruzkov functional method (see for instance [5][6][7][8][9][10][11][12][13][14][15][16][17][18]). …”

Section: Introductionmentioning

confidence: 99%

“…Now if a is negative, then with w n j = f (x j− 1 2 , t n ), one gets again Z n j = 0. From these observations, let us write with z j defined by (12) …”

Section: Introduction Of a Point Where The Error Is Estimatedmentioning

confidence: 99%

Error estimate for the upwind finite volume method for the nonlinear scalar conservation law

Bouche

Ghidaglia

Pascal

2011

Journal of Computational and Applied Mathematics

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MSC: 65M08 65M12 65M15 76M12 Keywords:Finite volume method Linear and nonlinear scalar problem Stability and convergence of numerical methods Geometric corrector a b s t r a c tIn this paper we estimate the error of upwind first order finite volume schemes applied to scalar conservation laws. As a first step, we consider standard upwind and flux finite volume scheme discretization of a linear equation with space variable coefficients in conservation form. We prove that, in spite of their lack of consistency, both schemes lead to a first order error estimate. As a final step, we prove a similar estimate for the nonlinear case. Our proofs rely on the notion of geometric corrector, introduced in our previous paper by [24] in the context of constant coefficient linear advection equations.

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Error Estimates for Finite Volume Approximations of Classical Solutions for Nonlinear Systems of Hyperbolic Balance Laws

Cited by 24 publications

References 28 publications

Convergence of Finite Volume Schemes for the Euler Equations via Dissipative Measure-Valued Solutions

Convergence of Finite Volume Schemes for the Euler Equations via Dissipative Measure-Valued Solutions

A Hyperbolic–Elliptic Model Problem for Coupled Surface–Subsurface Flow

Error estimate for the upwind finite volume method for the nonlinear scalar conservation law

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