High order finite volume schemes  based on reconstruction of states  for  solving  hyperbolic systems  with nonconservative products.   Applications to shallow-water systems

Castro, Manuel J.; Gallardo, José M.; Parés, Carlos

doi:10.1090/s0025-5718-06-01851-5

Cited by 258 publications

(260 citation statements)

References 34 publications

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“…Together with the first order schemes presented here, we also consider their high order extensions based on the use of a reconstruction operator following the ideas given in [7], [20]. As expected, the numerical results given by the first order numerical schemes introduced here are worse than those given by Roe methods, but when they are extended to higher order, the quality of the results is similar.…”

Section: Introductionmentioning

confidence: 71%

“…The numerical schemes have been extended to high order by following the ideas developed in [7,20,12] by means of the use of the PHM third order reconstruction operator (see [18]). …”

Section: Discussionmentioning

confidence: 99%

“…Following [7,20,12] we have also derived some third order extensions of all the first order numerical schemes presented here. These extensions are based on the third order reconstruction operator PHM (piecewise hyperbolic method) introduced in [18].…”

Section: Numerical Testsmentioning

confidence: 99%

See 2 more Smart Citations

On some fast well-balanced first order solvers for nonconservative systems

Castro¹,

Pardo²,

Parés³

et al. 2009

Math. Comp.

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Abstract. The goal of this article is to design robust and simple first order explicit solvers for one-dimensional nonconservative hyperbolic systems. These solvers are intended to be used as the basis for higher order methods for one or multidimensional problems. The starting point for the development of these solvers is the general definition of a Roe linearization introduced by Toumi in 1992 based on the use of a family of paths. Using this concept, Roe methods can be extended to nonconservative systems. These methods have good wellbalanced and robustness properties, but they have also some drawbacks: in particular, their implementation requires the explicit knowledge of the eigenstructure of the intermediate matrices. Our goal here is to design numerical methods based on a Roe linearization which overcome this drawback. The idea is to split the Roe matrices into two parts which are used to calculate the contributions at the cells to the right and to the left, respectively. This strategy is used to generate two different one-parameter families of schemes which contain, as particular cases, some generalizations to nonconservative systems of the well-known Lax-Friedrichs, Lax-Wendroff, FORCE, and GFORCE schemes. Some numerical experiments are presented to compare the behaviors of the schemes introduced here with Roe methods.

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Section: Introductionmentioning

confidence: 71%

“…The numerical schemes have been extended to high order by following the ideas developed in [7,20,12] by means of the use of the PHM third order reconstruction operator (see [18]). …”

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

On some fast well-balanced first order solvers for nonconservative systems

Castro¹,

Pardo²,

Parés³

et al. 2009

Math. Comp.

Self Cite

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show abstract

“…This additional complexity may require specialized numerical treatment in order to converge to appropriately defined weak solutions (e.g. [61]). …”

Section: Appendix a Correspondence Of Model Equationsmentioning

confidence: 99%

A depth-averaged debris-flow model that includes the effects of evolving dilatancy. II. Numerical predictions and experimental tests

2014

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We evaluate a new depth-averaged mathematical model that is designed to simulate all stages of debris-flow motion, from initiation to deposition. A companion paper shows how the model's five governing equations describe simultaneous evolution of flow thickness, solid volume fraction, basal pore-fluid pressure and two components of flow momentum. Each equation contains a source term that represents the influence of state-dependent granular dilatancy. Here, we recapitulate the equations and analyse their eigenstructure to show that they form a hyperbolic system with desirable stability properties.To solve the equations, we use a shock-capturing numerical scheme with adaptive mesh refinement, implemented in an open-source software package we call D-Claw. As tests of D-Claw, we compare model output with results from two sets of largescale debris-flow experiments. One set focuses on flow initiation from landslides triggered by rising pore-water pressures, and the other focuses on downstream flow dynamics, runout and deposition. D-Claw performs well in predicting evolution of flow speeds, thicknesses and basal pore-fluid pressures measured in each type of experiment. Computational results illustrate the critical role of dilatancy in linking coevolution of the solid volume fraction and porefluid pressure, which mediates basal Coulomb friction and thereby regulates debris-flow dynamics.

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“…Therefore, the adoption of well-balanced numerical methods (which are able to resolve quasi-steady problems) become an interesting option. In the case of shallow water flows with moving water, different well-balanced methods have been developed [4,5], and specific formulations have been presented for hyperbolic system of balance laws in the presence of stiff source terms [6] as well as multiscale phenomena [7]. However, in the present work we shall restrict our analysis to well-balanced schemes in the sense described in [8,9].…”

Section: Introductionmentioning

confidence: 99%

Hydrodynamic Instabilities in Well-Balanced Finite Volume Schemes for Frictional Shallow Water Equations. The Kinematic Wave Case

Bohorquez

Rentschler

2010

J Sci Comput

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We report the developments of hydrodynamic instabilities in several wellbalanced finite volume schemes that are observed during the computation of the temporal evolution of an out-balance flow which is essentially a kinematic wave. The numerical simulations are based on the one-dimensional shallow-water equations for a uniformly sloping bed with hydraulic resistance. Subsequently, we highlight the need of low dissipative high-order well-balanced filter schemes for non-equilibrium flows with variable cut-off wavenumber to compute the out-balance flow under consideration, i.e. the kinematic wave.

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High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems

Cited by 258 publications

References 34 publications

On some fast well-balanced first order solvers for nonconservative systems

On some fast well-balanced first order solvers for nonconservative systems

A depth-averaged debris-flow model that includes the effects of evolving dilatancy. II. Numerical predictions and experimental tests

Hydrodynamic Instabilities in Well-Balanced Finite Volume Schemes for Frictional Shallow Water Equations. The Kinematic Wave Case

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