Energy balanced numerical schemes with very high order. The Augmented Roe Flux ADER scheme. Application to the shallow water equations

Navas-Montilla, Adrián; Murillo, J.

doi:10.1016/j.jcp.2015.03.002

Cited by 28 publications

(31 citation statements)

References 53 publications

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“…In this case, the exact solution, in absence of discontinuities of the conservative variables (i.e. in absence of bores), is characterized by the constancy of the discharge and of the total head [6,7]. For the latter property of the exact steady solution, a numerical scheme that is able to preserve an initial steady state is defined energy balanced in [6].…”

Section: Introductionmentioning

confidence: 99%

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A comparison between bottom-discontinuity numerical treatments in the DG framework

Caleffi

Valiani

2016

Applied Mathematical Modelling

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In this work, using an unified framework consisting in third-order accurate discontinuous Galerkin schemes, we perform a comparison between five different numerical approaches to the free-surface shallow flow simulation on bottom steps. \ud \ud Together with the study of the overall impact that such techniques have on the numerical models we highlight the role that the treatment of bottom discontinuities plays in the preservation of specific asymptotic conditions. In particular, we consider three widespread approaches that perform well if the motionless steady state has to be preserved and two approaches (one previously conceived by the first two authors and one original) which are also promising for the preservation of a moving-water steady state.\ud \ud Several one-dimensional test cases are used to verify the third-order accuracy of the models in simulating an unsteady flow, the behavior of the models for a quiescent flow in the cases of both continuous and discontinuous bottom, and the good resolution properties of the schemes. Moreover, specific test cases are introduced to show the behavior of the different approaches when a bottom step interact with both steady and unsteady moving flows

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Section: Introductionmentioning

confidence: 99%

“…In this work, we have preferred to follow the idea presented in [16,17,15,6,7] for its internal consistency with the mathematical properties of the SWE.…”

Section: Introductionmentioning

confidence: 99%

A comparison between bottom-discontinuity numerical treatments in the DG framework

Caleffi

Valiani

2016

Applied Mathematical Modelling

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“…The system of Eqs. (11), (16), (17), (18) in the four unknowns h 1 , u 1 , h 2 , u 2 is solved using the Newton-Raphson method. Then, the downstream shock celerity c 2 is found using Eq.…”

Section: Expansion Large Depth Ratiomentioning

confidence: 99%

“…Successful applications showing energy conservation at geometric singularities can also be found in [15,16,17]. The well-balancing process and an extensive comparison between different numerical approaches of the free surface flow over bottom discontinuities are discussed in [18,19].…”

Section: Introductionmentioning

confidence: 99%

Dam break in rectangular channels with different upstream-downstream widths

Valiani

Caleffi

2019

Advances in Water Resources

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The classic Stoker dam-break problem [1] is revisited in cases of different channel widths upstream and downstream of the dam. The channel is supposed to have a rectangular cross section and a horizontal and frictionless bottom. The system of the shallow water equations is enriched, using the width as a space-dependent variable, together with the depth and the unit discharge, which conversely depend on both space and time. Such a formulation allows a quasi-analytical treatment of the system, whose solution is similar to that of the classic Stoker solution when the downstream/upstream depth ratio is sufficiently large, except that a further stationary contact wave exists at the dam position. When the downstream/upstream depth ratio is small, the solution is richer than the Stoker solution because the critical state occurs at the dam position and the solution itself becomes resonant at the same position, where two eigenvalues are null and the strict hyperbolicity of the system is lost. The limits that identify the flow regime for channel contraction and channel expansion are discussed after showing that the nondimensional parameters governing the problem are the downstream/upstream width ratio and the downstream/upstream initial depth ratio.After the introduction of the previous analytical framework, a numerical analysis is also performed to evaluate a numerical method that is conceived to suitably capture rarefactions, shock waves and contact waves. A secondorder method is adopted, employing a Dumbser-Osher-Toro Riemann solver equipped with a nonlinear path. Such an original nonlinear path is shown to perform better than the classic linear path when contact waves of large amplitude must be captured, being able to obtain specific energy conservation and mass conservation at the singularity.The codes, written in MATLAB (MathWorks Inc.) language, are made available in Mendeley Data repository.

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“…The conservation of the discharge and of the total head are successfully applied in 1D modelling of open-channel hydraulics, both for the unit width channel and in a channel of arbitrary shape ( [9,10,11]).…”

Section: Introductionmentioning

confidence: 99%

Momentum balance in the shallow water equations on bottom discontinuities

Valiani¹,

Caleffi²

2017

Advances in Water Resources

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This work investigates the topical problem of balancing the shallow water equations over bottom steps of different heights. The current approaches in the literature are essentially based on mathematical analysis of the hyperbolic system of balance equations and take into account the relevant progresses in treating the non-conservative form of the governing system in the framework of path-conservative schemes. An important problem under debate is the correct position of the momentum balance closure when the bottom elevation is discontinuous. Cases of technical interest are systematically analysed, consisting of backward-facing steps and forward-facing steps, tackled supercritical and subcritical flows; critical (sonic) conditions are also analysed and discussed. The fundamental concept governing the problem and supported by the present computations is that the energy-conserving approach is the only approach that is consistent with the classical shallow water equations formulated with geometrical source terms and that the momentum balance is properly closed if a proper choice of a conventional depth on the bottom step is performed. The depth on the step is shown to be included between the depths just upstream and just downstream of the step. It is also shown that current choices (as given in the literature) of the depth on (or in front of) the step can lead to unphysical configurations, similar to some energy-increasing solutions

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Energy balanced numerical schemes with very high order. The Augmented Roe Flux ADER scheme. Application to the shallow water equations

Cited by 28 publications

References 53 publications

A comparison between bottom-discontinuity numerical treatments in the DG framework

A comparison between bottom-discontinuity numerical treatments in the DG framework

Dam break in rectangular channels with different upstream-downstream widths

Momentum balance in the shallow water equations on bottom discontinuities

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