Accuracy of the weighted essentially non-oscillatory conservative finite difference schemes

Don, Wai Sun; Borges, Rafael Brandão de Rezende

doi:10.1016/j.jcp.2013.05.018

Cited by 184 publications

(173 citation statements)

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“…In this paper we use the WENO-Z method of Don and Borges [17], which we summarize here for the case p = 5. At grid cell interfaces in the x-direction, edge averaged values of the conserved quantities are computed…”

Section: The Dimension-by-dimension Weno Methodsmentioning

confidence: 99%

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Finite volume WENO methods for hyperbolic conservation laws on Cartesian grids with adaptive mesh refinement

Buchmüller

Dreher

Helzel

2016

Applied Mathematics and Computation

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Section: The Dimension-by-dimension Weno Methodsmentioning

confidence: 99%

“…Formulas for the case p = 7 are summarized in Appendix A. Numerous one-dimensional test computations with the WENO-Z method can be found in [17]. These edge averaged interface values of the conserved quantities can now be used to compute interface fluxes.…”

Section: The Dimension-by-dimension Weno Methodsmentioning

confidence: 99%

Finite volume WENO methods for hyperbolic conservation laws on Cartesian grids with adaptive mesh refinement

Buchmüller

Dreher

Helzel

2016

Applied Mathematics and Computation

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“…In [4], the WENO-Z5 scheme has been shown, in comparison to the WENO-JS5 scheme, to allow a more flexible choice on the as a function of grid spacing x that will guarantee the formal order of convergence of the nonlinear scheme regardless of the critical points while maintaining the essentially non-oscillatory nature of the shock-capturing scheme. Readers are referred to [2][3][4] for more details.…”

Section: Remarkmentioning

confidence: 99%

A Spectral Study on the Dissipation and Dispersion of the WENO Schemes

2014

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The dissipation and dispersion (spectral) properties of the nonlinear fifth order classical weighted essentially non-oscillatory finite difference scheme (WENO-JS5) and its improved version (WENO-Z5) using the approximate dispersion relation (ADR) (Pirozzoli in J Comput Phys 219: [489][490][491][492][493][494][495][496][497] 2006) and the nonlinear spectral analysis (NSA) (Fauconnier et al. in J Comput Phys 228(6):1830-1861, 2009) are studied. Unlike the previous studies, the influences of the sensitivity parameter in the definition of the WENO nonlinear weights are also included for completeness. The fifth order upwinded central linear scheme (UW5) serves as the reference and benchmark for the purpose of comparison. The spectral properties of the WENO differentiation operator is well predicted theoretically by the ADR and validated numerically by the simulations of the WENO schemes in solving the scalar linear advection equation. In a long time simulation with an initial broadband wave, the WENO schemes generate spurious high modes with amplitude and spread of wavenumbers depend on the value of the sensitivity parameter. The NSA is applied to investigate the statistical nonlinear behavior, due to the nonlinear stencils adaptation of the WENO schemes, with a large set of initial conditions consisting of synthetic scalar fields with a prescribed energy spectrum and random phases. The statistics indicate that there is a small probability of an existence of a mild anti-dissipation in the low wavenumber range regardless of the size of the sensitivity parameter. Numerical examples demonstrate that the WENO-Z5 scheme is not only less dissipative and dispersive but also less sensitive to random phases than the WENO-JS5 scheme. Furthermore, a sensitivity parameter adaptive technique, in which its value depends 123 J Sci Comput on the local smoothness of the solution at a given spatial location and time, is introduced for solving a linear advection problem with a discontinuous initial condition. The preliminary result shows that the solution computed by the sensitivity parameter adaptive WENO-Z5 scheme agrees well with those computed by the WENO-Z5 scheme and the UW5 scheme in regions containing discontinuities and smooth solutions, respectively.

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“…More information about this two techniques can be found in scientific literature [54]. Additionally, these methods are being continuously researched and new improvements are presented for hyperbolic conservation laws [55][56][57].…”

Section: Enhancing Of the Derivatives Approximationsmentioning

confidence: 99%

“…On the other hand, when φ is a SDF, the condition |φ| = 1 is fulfilled and thus, the curvature can be approximated by the Laplacian of φ, therefore, the equality (2.36) is satisfied, leading to the heat equation: 56) where φ is the temperature and b is the thermal conductivity. The heat equation is the most basic parabolic equation and these equations need to be discretized using central differencing since the information to update the front is taken from all spatial directions, as opposed to hyperbolic equations where information flows only in the direction of characteristics.…”

Section: Dependent On Local Curvaturementioning

confidence: 99%

Study, Modelling and Implementation of the Level Set Method Used in Micromachining Processes

Álvaro¹

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"Sí com lo taur se'n va fuyt pel desert quantés sobrat per son semblant qui·l força, ne torna may fins ha cobrada força per destruir aquell qui l'ha desert: tot enaxí·m cové lunyar de vós, car vostre gest mon esforç ha confús: no tornaré fins del tot haja fus la gran pahor qui·m toll ser delitós." Ausiàs March, 1397 - †1459 Per a Marc AgradecimientosSin duda, todo el trabajo que hay detrás de esta tesis ha sido posible gracias a la colaboración y dedicación de muchos. Todos ellos merecen ser agradecidos por haber contribuido a su manera en la culminación de este trabajo.Agradecer sinceramente a mis directores y tutores, Ximo Cerdà y Ricardo Colom, por permitir formarme como investigador y ofrecerme sus consejos. Ellos me han tratado con respeto y nunca han puesto freno a mis ambiciones, más bien las han engrandecido. Gracias por guiar toda esta aventura.Una mención muy especial se merecen MiguelÁngel Gosálvez y Néstor Ferrando. Les tengo que agradecer la gran suerte que he tenido de poder colaborar con dos eminencias como ellos. Sus grandes conocimientos han permitido obtener los mejores resultados de este trabajo, además de haber sido una experiencia gratificante. Gracias por su constante apoyo y su ayuda infinita.Asimismo, dar las gracias también a Vicente Herrero, Jose María Monzó, Ana Ros y Ramón J. Aliaga. Ellos dieron vida a un ambiente de trabajo idóneo, aportándome mucho más que ayuda y conocimientos. Del mismo modoÁngel Tébar, Rafael Gadea y Jorge Daniel siempre han estado dispuestos a ayudar en todo cuanto han podido.Mi estancia en el instituto Fraunhofer de Erlangen me ha permitido acelerar mi formación y vivir unas experiencias de lo más satisfactorias. Todo ello ha sido gracias a la magnífica gente con la que he tenido el placer de colaborar y convivir. Gracias a Eberhard Bär y a toda la incontable gente del Fraunhofer IISB.Un mérito especial se merecen mi familia y mis amigos por aguantarme todo este tiempo sin perder la paciencia conmigo ni en los peores momentos. Es de un valor incalculable tener tantos hombros en los que poder dejarte caer para coger las fuerzas suficientes para seguir. Gracias Isidor y Lourdes, gracias Ferran y Andreu, y gracias Vicente, Enric y David.iii No podía faltar mi particular aimia, Nuria.Ella se merece un eterno agradecimiento por haber tenido que vivir todo esto conmigo. Compañera de todas y cada una de las experiencias, buenas y malas, de esta larga aventura. Animándome en todo momento, siempre ayudándome con un altruismo deslumbrante y enseñándome una inmensidad de conocimientos. Ella, ha sido y es, mi llir entre cards.Una vez más, muchísimas gracias a todos. iv SummaryThe main topic of the present thesis is the improvement of fabrication processes simulation by means of the Level Set (LS) method. The LS is a mathematical approach used for evolving fronts according to a motion defined by certain laws. The main advantage of this method is that the front is embedded inside a higher dimensional function such that updating this function instead of directly the front it...

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Accuracy of the weighted essentially non-oscillatory conservative finite difference schemes

Cited by 184 publications

References 18 publications

Finite volume WENO methods for hyperbolic conservation laws on Cartesian grids with adaptive mesh refinement

Finite volume WENO methods for hyperbolic conservation laws on Cartesian grids with adaptive mesh refinement

A Spectral Study on the Dissipation and Dispersion of the WENO Schemes

Study, Modelling and Implementation of the Level Set Method Used in Micromachining Processes

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