Adaptive mesh reconstruction for hyperbolic conservation laws with total variation bound

Sfakianakis, Nikolaos

doi:10.1090/s0025-5718-2012-02615-9

Cited by 7 publications

(13 citation statements)

References 17 publications

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“…The approach that we follow, for the mesh reconstruction step of MAS (Step 1) was first introduced by Arvanitis, Katsaounis & Makridakis (2001) and by Arvanitis (2002). Applications of MAS on several problems, point out a strong stabilisation property emanating from the mesh reconstruction Arvanitis, Makridakis & Tzavaras (2004), Arvanitis & Delis (2006), Arvanitis, Makridakis & Sfakianakis (2006), Sfakianakis (2011), Sfakianakis (2009. These stabilization properties led Arvanitis, Makridakis & Sfakianakis (2006) to combine MAS with the marginal class of entropy conservative schemes.…”

Section: Introductionmentioning

confidence: 99%

Entropy dissipation of moving mesh adaptation

Lukacova-Medvid

Sfakianakis

2014

J. Hyper. Differential Equations

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Non-uniform grids and mesh adaptation have been a growing part of numerical simulation over the past years. It has been experimentally noted that mesh adaptation leads not only to locally improved solution but also to numerical stability of the underlying method. There have been though only few results on the mathematical analysis of these schemes (see for example Sfakianakis (2011)) due to the lack of proper tools that incorporate both the time evolution and the mesh adaptation step of the overall algorithm.In this paper we provide a method to perform the analysis of the mesh adaptation method, including both the mesh reconstruction and evolution of the solution. We moreover employ this method to extract sufficient conditions -on the adaptation of the mesh-that stabilize a numerical scheme in the sense of the entropy dissipation. arXiv:1209.5009v1 [math.NA]

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Section: Introductionmentioning

confidence: 99%

Entropy dissipation of moving mesh adaptation

Lukacova-Medvid

Sfakianakis

2014

J. Hyper. Differential Equations

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“…Applications of the main adaptive scheme on several problems, point out a strong stabilization property emanating from the mesh reconstruction [1,3,14]. This stabilization property led us to combine MAS with the marginal class of entropy conservative schemes.…”

Section: Introductionmentioning

confidence: 99%

Entropy Conservative Schemes and Adaptive Mesh Selection for Hyperbolic Conservation Laws

Arvanitis

Makridakis

Sfakianakis

2010

J. Hyper. Differential Equations

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We consider numerical schemes which combine non-uniform, adaptively redefined spatial meshes with entropy conservative schemes for the evolution step for shock computations. We observe that the resulting adaptive schemes yield approximations free of oscillations in contrast to known fully discrete entropy conservative schemes on uniform meshes. We conclude that entropy conservative schemes are transformed to entropy diminishing schemes when combined with the proposed geometrically driven mesh adaptivity.with initial data u 0 having compact support in a much smaller interval.We are interested in studying numerical approximations of the exact solution u of (1.1) over non-uniform, adaptively redefined spatial meshes. Adaptivity is a main theme in modern scientific computing of complex physical phenomena. It is therefore important to investigate the behavior of adaptive schemes for hyperbolic 383 J. Hyper. Differential Equations 2010.07:383-404. Downloaded from www.worldscientific.com by NEW YORK UNIVERSITY on 02/08/15. For personal use only.

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“…This technique has been used by many authors in past and also been analyzed for several mathematical properties like TVB and maximum principle. One key feature of r-refinement technique is no addition of extra-point in the discretized domain see references [1,24,7,21,22]. These rrefinement techniques rely on the choice of monitor and estimator function based on equi-distribution principle.…”

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“…In [21], a moving mesh algorithm is proposed for scalar one dimensional conservation law using curvature as an estimator function. Various important properties such as TVB, maximum principle are also shown for finite volume numerical solution obtained by the moving mesh algorithm [21]. In [2], an entropy conservative scheme combined with the MAS algorithm was designed and shown the entropy dissipation of numerical solution.…”

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A mesh adaptation algorithm using new monitor and estimator function for discontinuous and layered solution

Mishra¹,

Gupta²,

Dubey³

2022

NACO

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<p style='text-indent:20px;'>In this work a novel mesh adaptation technique is proposed to approximate discontinuous or boundary layer solution of partial differential equations. We introduce new estimator and monitor function to detect solution region containing discontinuity and layered region. Subsequently, this information is utilized along with equi-distribution principle to adapt the mesh locally. Numerical tests for numerous scalar problems are presented. These results clearly demonstrate the robustness of this method and non-oscillatory nature of the computed solutions.</p>

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Adaptive mesh reconstruction for hyperbolic conservation laws with total variation bound

Cited by 7 publications

References 17 publications

Entropy dissipation of moving mesh adaptation

Entropy dissipation of moving mesh adaptation

Entropy Conservative Schemes and Adaptive Mesh Selection for Hyperbolic Conservation Laws

A mesh adaptation algorithm using new monitor and estimator function for discontinuous and layered solution

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