A posteriori error estimation for finite-volume solutions of hyperbolic conservation laws

Zhang, X.D.; Trépanier, Jean‐Yves; Camarero, R.

doi:10.1016/s0045-7825(99)00099-7

Cited by 73 publications

(53 citation statements)

References 14 publications

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“…Many works studying the a posteriori error estimate base on mathematical arguments [28]. But paradoxically, to our knowledge, very few works use a physical criterion [5]  the numerical density of entropy production.…”

Section: Mesh Refinement Criterionmentioning

confidence: 99%

Adaptive multiscale scheme based on numerical density of entropy production for conservation laws

Ersoy¹,

Golay²,

Yushchenko³

2013

Open Mathematics

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We propose a 1D adaptive numerical scheme for hyperbolic conservation laws based on the numerical density of entropy production (the amount of violation of the theoretical entropy inequality). This density is used as an a posteriori error which provides information if the mesh should be refined in the regions where discontinuities occur or coarsened in the regions where the solution remains smooth. As due to the Courant-Friedrich-Levy stability condition the time step is restricted and leads to time consuming simulations, we propose a local time stepping algorithm. We also use high order time extensions applying the Adams-Bashforth time integration technique as well as the second order linear reconstruction in space. We numerically investigate the efficiency of the scheme through several test cases: Sod's shock tube problem, Lax's shock tube problem and the Shu-Osher test problem. MSC:

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Section: Mesh Refinement Criterionmentioning

confidence: 99%

Adaptive multiscale scheme based on numerical density of entropy production for conservation laws

Ersoy¹,

Golay²,

Yushchenko³

2013

Open Mathematics

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“…The source of error is the residual of the approximate solution, and, in the nonlinear case, the error evolves by a different differential operator than the solution. In the linear error transport approach [10][11][12][13][14][15][16][17][18], the differential operator on the left-hand side of (4) is linearized aboutũ , which is a reasonable assumption so long as |e(x, t)| |u(x, t)| . In practice, and in some very important cases, this is not valid, and such a linearization becomes questionable.…”

Section: Basic Conceptsmentioning

confidence: 99%

“…Therefore, the use of the same orders for primal and error equation discretizations results in asymptotically correct error estimates so long as the residual approximation is sufficiently accurate, as demonstrated in [10,16]. The evaluation of the residual has been a primary focus of investigation in the literature to date.…”

Section: Basic Conceptsmentioning

confidence: 99%

“…Approximate the residual by a discretization; 2. Approximate the residual by one or more leading truncation error terms from the modified differential equation of the primal discretization [10,17]; 3. Approximate the residual by reconstructing the approximate solution and directly applying the differential residual operator [12,16,19,20].…”

Section: Basic Conceptsmentioning

confidence: 99%

“…In all cases in the error transport literature, the error equations are linearized, and the focus is typically on the approximation used to evaluate the local residual, that is, the local source (sink) of error that drives an error transport equation. Linear error transport techniques have been applied to linear and nonlinear models of advection [10,11] and advection-diffusion [11][12][13][14][15][16] equations, subsonic and supersonic inviscid flow [10,17,18], and subsonic viscous flow [12,16]. Most investigations have considered steady-state solutions [13][14][15][16][17][18], and only a few examples of timedependent applications [10,11] exist.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Numerical error estimation for nonlinear hyperbolic PDEs via nonlinear error transport

Banks

Hittinger

Connors

et al. 2012

Computer Methods in Applied Mechanics and Engineering

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The estimation of discretization error in numerical simulations is a key component in the development of uncertainty quantification. In particular, there exists a need for reliable, robust estimators for finite volume and finite difference discretizations of hyperbolic partial differential equations. The approach espoused here, often called the error transport approach in the literature, is to solve an auxiliary error equation concurrently with the primal governing equation to obtain a point-wise (cell-wise) estimate of the discretization error. One-dimensional, nonlinear, time-dependent problems are considered. In contrast to previous work, fully nonlinear error equations are advanced, and potential benefits are identified. A systematic approach to approximate the local residual for both method-of-lines and space-time discretizations is developed. Behavior of the error estimates on problems that include weak solutions demonstrates that this nonlinear error evolution provides useful error estimates.

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Numerical error patterns for a scheme with hermite interpolation for 1 + 1 linear wave equations

Zhu

Wang

2004

Commun. Numer. Meth. Engng.

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SUMMARYNumerical error patterns were presented when the fourth-order scheme based on Hermite interpolation was used to solve the 1 + 1 linear wave equation. Since most non-linear equations for real systems can be converted into linear forms by using proper transformations, this study certainly pertains its practical signiÿcance. The analytical solution was obtained under inhomogeneous initial and boundary conditions. It was found that not only the Hurst index of an error train at a given position but also its spatial distribution is dependent on the ratio of temporal to spatial intervals. The solution process with the fourth-order scheme based on Hermite interpolation diverges as the ratio is greater than unity. The results show that regular error pattern and smaller maxima of absolute values of numerical errors can be obtained when the ratio is set as unity; while chaotic phenomena for the numerical error propagation process can appear when the ratio is less than unity. It was found that it is better to choose the ratio as unity for the numerical solution of 1 + 1 linear wave equation with the scheme; while other selections for the ratio in the scheme can bring about chaotic patterns for the numerical errors.

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A posteriori error estimation for finite-volume solutions of hyperbolic conservation laws

Cited by 73 publications

References 14 publications

Adaptive multiscale scheme based on numerical density of entropy production for conservation laws

Adaptive multiscale scheme based on numerical density of entropy production for conservation laws

Numerical error estimation for nonlinear hyperbolic PDEs via nonlinear error transport

Numerical error patterns for a scheme with hermite interpolation for 1 + 1 linear wave equations

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