On the well-balance property of Roe's method for nonconservative hyperbolic systems. applications to shallow-water systems

Parés, Carlos; Castro, Manuel J.

doi:10.1051/m2an:2004041

Cited by 184 publications

(217 citation statements)

References 32 publications

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“…According to [21], we introduce the following definitions: Definition 2.1. Given a curve γ ∈ Γ, a numerical scheme for solving (2.1) is said to be exactly well-balanced for γ if it solves exactly any smooth stationary solution W such that (2.10)…”

Section: Preliminariesmentioning

confidence: 99%

“…It is a rather formal and geometrical definition, but it is well suited to the analysis of the well-balanced properties of path-conservative numerical schemes as these properties are strongly connected to the relationship between the paths and the set of curves Γ; see [20]. For instance, the following general result can be easily shown for Roe methods (see [21] The proof of this proposition is straightforward: on the one hand, it can be easily deduced from (2.7) that, given two states W L and W R belonging to γ, the following equality holds:…”

Section: Preliminariesmentioning

confidence: 99%

“…As in [21], we consider families of paths Φ which are an extension to Ω of a family of paths in O; i.e., we assume that there exists a family of paths φ :…”

Section: Systems Of Conservation Laws With Source Terms and Nonconsermentioning

confidence: 99%

“…Therefore, the Roe scheme is exactly well-balanced for water at rest or vacuum stationary solutions. Moreover, it is well-balanced of order 2 for general stationary solutions (see [21]). …”

Section: Shallow Water Equations With Depth Variationsmentioning

confidence: 99%

“…Although the schemes of this family are robust and have good well-balanced properties (see, for instance, [2], [8], [21], [20]) they also present, as their conservative counterpart, some drawbacks:…”

Section: Introductionmentioning

confidence: 99%

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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: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

“…As in [21], we consider families of paths Φ which are an extension to Ω of a family of paths in O; i.e., we assume that there exists a family of paths φ :…”

Section: Systems Of Conservation Laws With Source Terms and Nonconsermentioning

confidence: 99%

“…Therefore, the Roe scheme is exactly well-balanced for water at rest or vacuum stationary solutions. Moreover, it is well-balanced of order 2 for general stationary solutions (see [21]). …”

Section: Shallow Water Equations With Depth Variationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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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A well‐balanced FV scheme for compound channels with complex geometry and movable bed

Minatti

2015

Water Resources Research

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This work focuses on the implementation of a Shallow Water-Exner model for compound natural channels with complex geometry and movable bed within the finite volume framework. The model is devised for compound channels modeling: cross-section overbanks are treated with fixed bed conditions, while the main channel is left free to modify its morphology. A capacitive approach is used for bed load transport modeling, in which the solid flow rates are estimated with bed load transport formulas. The model equations pose some numerical issues in the case of natural channels, where bed load transport may occur for both subcritical and supercritical flows and geometry varies in space. An explicit path-conservative scheme, designed to overcome all these issues, is presented in the paper. The scheme solves liquid and solid phases dynamics in a coupled manner, in order to correctly model near critical currents/channel interactions and is well-balanced, that is able to properly reproduce steady states. The Roe and Osher Riemann solvers are implemented, so as to take into account the spatial geometry variations of natural channels. The scheme reaches up to second-order accuracy. Validation is performed with fixed and movable bed test cases whose analytical solution is known, and with flume experimental data. An application of the model to a real case study is also shown.

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Well‐balanced high‐order solver for blood flow in networks of vessels with variable properties

Müller

Toro

2013

Numer Methods Biomed Eng

100

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We present a well-balanced, high-order non-linear numerical scheme for solving a hyperbolic system that models one-dimensional flow in blood vessels with variable mechanical and geometrical properties along their length. Using a suitable set of test problems with exact solution, we rigorously assess the performance of the scheme. In particular, we assess the well-balanced property and the effective order of accuracy through an empirical convergence rate study. Schemes of up to fifth order of accuracy in both space and time are implemented and assessed. The numerical methodology is then extended to realistic networks of elastic vessels and is validated against published state-of-the-art numerical solutions and experimental measurements. It is envisaged that the present scheme will constitute the building block for a closed, global model for the human circulation system involving arteries, veins, capillaries and cerebrospinal fluid.

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On the well-balance property of Roe's method for nonconservative hyperbolic systems. applications to shallow-water systems

Cited by 184 publications

References 32 publications

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

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

A well‐balanced FV scheme for compound channels with complex geometry and movable bed

Well‐balanced high‐order solver for blood flow in networks of vessels with variable properties

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