A Technique of Treating Negative Weights in WENO Schemes

Shi, Ji‐Quan; Hu, Changqing; Shu, Chi‐Wang

doi:10.1006/jcph.2001.6892

Cited by 341 publications

(308 citation statements)

References 27 publications

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“…The external force term f represents the gravitational force only, neglecting surface tension forces and Coriolis. The advection term is discretized with the fifth-order WENO (weighted essentially non-oscillatory) scheme (Shi et al, 2002) or the second-order TVD (total variation diminishing) Superbee scheme (Darwish and Moukalled, 2003) in separate numerical tests. The diffusion term on the right-hand side of Eq.…”

Section: Governing Equationsmentioning

confidence: 99%

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Comparing thixotropic and Herschel–Bulkley parameterizations for continuum models of avalanches and subaqueous debris flows

Jeon

Hodges

2018

Nat. Hazards Earth Syst. Sci.

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Abstract. Avalanches and subaqueous debris flows are two cases of a wide range of natural hazards that have been previously modeled with non-Newtonian fluid mechanics approximating the interplay of forces associated with gravity flows of granular and solid-liquid mixtures. The complex behaviors of such flows at unsteady flow initiation (i.e., destruction of structural jamming) and flow stalling (restructuralization) imply that the representative viscositystress relationships should include hysteresis: there is no reason to expect the timescale of microstructure destruction is the same as the timescale of restructuralization. The nonNewtonian Herschel-Bulkley relationship that has been previously used in such models implies complete reversibility of the stress-strain relationship and thus cannot correctly represent unsteady phases. In contrast, a thixotropic nonNewtonian model allows representation of initial structural jamming and aging effects that provide hysteresis in the stress-strain relationship. In this study, a thixotropic model and a Herschel-Bulkley model are compared to each other and to prior laboratory experiments that are representative of an avalanche and a subaqueous debris flow. A numerical solver using a multi-material level-set method is applied to track multiple interfaces simultaneously in the simulations. The numerical results are validated with analytical solutions and available experimental data using parameters selected based on the experimental setup and without post hoc calibration. The thixotropic (time-dependent) fluid model shows reasonable agreement with all the experimental data. For most of the experimental conditions, the HerschelBulkley (time-independent) model results were similar to the thixotropic model, a critical exception being conditions with a high yield stress where the Herschel-Bulkley model did not initiate flow. These results indicate that the thixotropic relationship is promising for modeling unsteady phases of debris flows and avalanches, but there is a need for better understanding of the correct material parameters and parameters for the initial structural jamming and characteristic time of aging, which requires more detailed experimental data than presently available.

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Section: Governing Equationsmentioning

confidence: 99%

“…The multi-material level-set method herein follows Merriman et al (1994) with the addition of high-order numerical schemes (Shu and Osher, 1989;Shi et al, 2002). The "level set" of the ith fluid is designated as φ i :…”

Section: Estimation Of Parameters For Time-dependent Coussot Modelmentioning

confidence: 99%

Comparing thixotropic and Herschel–Bulkley parameterizations for continuum models of avalanches and subaqueous debris flows

Jeon

Hodges

2018

Nat. Hazards Earth Syst. Sci.

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“…Explicit formulas for WENO coefficients on uniform meshes appear in, e.g., [7,8]. A framework for deriving WENO coefficients for non-uniform meshes is established in [7]; explicit formulas appear for uniform meshes.…”

Section: Introductionmentioning

confidence: 99%

“…WENO coefficients on arbitrary triangular meshes are derived for second-order schemes in [14] and for third-and fourth-order schemes in [15]. Computations in two dimensions with WENO discretizations are performed in [8] on triangular and rectangular meshes.…”

Section: Introductionmentioning

confidence: 99%

Observations on the fifth-order WENO method with non-uniform meshes

Wang

Feng

Spiteri

2008

Applied Mathematics and Computation

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The weighted essentially non-oscillatory (WENO) methods are a popular high-order spatial discretization for hyperbolic partial differential equations. Typical treatments of WENO methods assume a uniform mesh. In this paper we give explicit formulas for the finite-volume, fifth-order WENO (WENO5) method on non-uniform meshes in a way that is amenable to efficient implementation. We then compare the performance of the non-uniform mesh approach with the classical uniform mesh approach for the finite-volume formulation of the WENO5 method. We find that the numerical results significantly favor the non-uniform mesh approach both in terms of computational efficiency as well as memory usage. We expect this investigation to provide a basis for future work on adaptive mesh methods coupled with the finite-volume WENO methods.

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“…Other first and second order schemes have been applied to the SWE using the idea of balancing the source term and the flux gradients, for example the wave propagation algorithm by LeVeque [46], the kinetic scheme by Xu [62], and Perthame and Simeoni [48], central-upwind schemes proposed by Kurganov and Levy [39], and a family of flux-splitting numerical solvers proposed in [51]. It is worth pointing out that the contributions mentioned above, devoted to the numerical solution of hyperbolic problems with stiff source terms, do not address in general the question of convergence of the method, in particular when the system is singular.…”

Section: Introductionmentioning

confidence: 99%

A sign matrix based scheme for non-homogeneous PDE’s with an analysis of the convergence stagnation phenomenon

Sahmim¹,

Benkhaldoun

Alcrudo

2007

Journal of Computational Physics

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To cite this version:Slah Sahmim, Fayssal Benkhaldoun, Francisco Alcrudo. A sign matrix based scheme for nonhomogeneous PDE's with an analysis of the convergence stagnation phenomenon. Journal of Computational Physics, Elsevier, 2007, 226 (2) AbstractThis work is devoted to the analysis of a finite volume method recently proposed for the numerical computation of a class of non homogenous systems of partial differencial equations of interest in fluid dynamics. The stability analysis of the proposed scheme leads to the introduction of the sign matrix of the flux jacobian. It appears that this formulation is equivalent to the VFRoe scheme introduced in the homogeneous case and has a natural extension here to non homogeneous systems. Comparative numerical experiments for the Shallow Water and Euler equations with source terms, and a model problem of two phase flow (Ransom faucet) are presented to validate the scheme. The numerical results present a convergence stagnation phenomenon for certain forms of the source term, notably when it is singular. Convergence stagnation has been also shown in the past for other numerical schemes. This issue is addressed in a specific section where an explanation is given with the help of a linear model equation, and a cure is demonstrated.

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A Technique of Treating Negative Weights in WENO Schemes

Cited by 341 publications

References 27 publications

Comparing thixotropic and Herschel–Bulkley parameterizations for continuum models of avalanches and subaqueous debris flows

Comparing thixotropic and Herschel–Bulkley parameterizations for continuum models of avalanches and subaqueous debris flows

Observations on the fifth-order WENO method with non-uniform meshes

A sign matrix based scheme for non-homogeneous PDE’s with an analysis of the convergence stagnation phenomenon

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