An efficient hyperbolic relaxation system for dispersive non-hydrostatic water waves and its solution with high order discontinuous Galerkin schemes

Escalante, Cipriano; Dumbser, Michael; Castro, Manuel J.

doi:10.1016/j.jcp.2019.05.035

Cited by 37 publications

(68 citation statements)

References 110 publications

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“…As pointed out in [17], [18], [21] among others, in shallow water flow complex events can be observed related to turbulent processes. One of these processes corresponds to the breaking of waves near the coast.…”

Section: Modeling Of Breaking Wavesmentioning

confidence: 80%

“…So that the general formulation (9) is reduced to (12) in this situation. This means that, for the case of flat bottom, this system is the same as (BMSS) but setting p b = 3 2 p. A particular analysis under this hypothesis of flat bottom, as well as an efficient numerical strategy to solve it was proposed in [13] and [18] respectively.…”

Section: Rewriting Of Classical Boussinesq and Non-hydrostatic Pressumentioning

confidence: 99%

“…As it will be seen in the numerical tests shown in the next section, the PDE systems considered in this paper cannot describe this process without an additional term that allows the model to dissipate the required amount of energy in such situations. In this work, we adopt a simplified version of the breaking mechanism introduced in [17], [22] and later adopted for approximated hyperbolic dispersive systems in [18] where a friction term is added to the right hand side of the equations:…”

Section: Modeling Of Breaking Wavesmentioning

confidence: 99%

“…The second goal of the paper will be to use the proposed general non-hydrostatic system to present a general numerical approach. The approach is based on a technique similar to the one introduced in [18]. The idea is to propose a relaxation of the system which will cover the original one in the relaxed limit.…”

Section: Introductionmentioning

confidence: 99%

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A General Non-hydrostatic Hyperbolic Formulation for Boussinesq Dispersive Shallow Flows and Its Numerical Approximation

Escalante

Luna

2020

J Sci Comput

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In this paper, we propose a novel first-order reformulation of the most well-known Boussinesq-type systems that are used in ocean engineering. This has the advantage of collecting in a general framework many of the well-known systems used for dispersive flows. Moreover, it avoids the use of high-order derivatives which are not easy to treat numerically, due to the large stencil usually needed. These first-order PDE dispersive systems are then approximated by a novel set of first-order hyperbolic equations. Our new hyperbolic approximation is based on a relaxed augmented system in which the divergence constraints of the velocity flow variables are coupled with the other conservation laws via an evolution equation for the depth-averaged non-hydrostatic pressures. The most important advantage of this new hyperbolic formulation is that it can be easily discretized with explicit and high-order accurate numerical schemes for hyperbolic conservation laws. There is no longer need of solving implicitly some linear system as it is usually done in many classical approaches of Boussinesq-type models. Here a third-order finite volume scheme based on a CWENO reconstruction has been used. The scheme is well-balanced and can treat correctly wet-dry areas and emerging topographies. Several numerical tests, which include idealized academic benchmarks and laboratory experiments are proposed, showing the advantage, efficiency and accuracy of the technique proposed here.

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Section: Modeling Of Breaking Wavesmentioning

confidence: 80%

Section: Rewriting Of Classical Boussinesq and Non-hydrostatic Pressumentioning

confidence: 99%

Section: Modeling Of Breaking Wavesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A General Non-hydrostatic Hyperbolic Formulation for Boussinesq Dispersive Shallow Flows and Its Numerical Approximation

Escalante

Luna

2020

J Sci Comput

Self Cite

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“…Different from classical a priori limiting schemes, the Multi-dimensional Optimal Order Detection (MOOD), also known as a posteriori limiting scheme, was proposed in [10,19,21], and has been employed in various numerical formulations [7,20,6] and applications [11,22,8,36,4,24,61,56] and more recently [23,57,27]. In MOOD algorithm, some criterions or detectors were designed based on the computational stability and physical properties.…”

Section: Introductionmentioning

confidence: 99%

Solution property preserving reconstruction BVD+MOOD scheme for compressible euler equations with source terms and detonations

et al. 2020

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In "Solution Property Reconstruction for Finite Volume scheme: a BVD+MOOD framework", Int. J. Numer. Methods Fluids, 2020, we have designed a novel solution property preserving reconstruction, so-called multi-stage BVD-MOOD scheme. The scheme is able to maintain a high accuracy in smooth profile, eliminate the oscillations in the vicinity of discontinuity, capture sharply discontinuity and preserve some physical properties like the positivity of density and pressure for the Euler equations of compressible gas dynamics. In this paper, we present an extension of this approach for the compressible Euler equations supplemented with source terms (e.g., gravity, chemical reaction). One of the main challenges when simulating these models is the occurrence of negative density or pressure during the time evolution, which leads to a blow-up of the computation. General compressible Euler equations with different type of source terms are considered as models for physical situations such as detonation waves. Then, we illustrate the performance of the proposed scheme via a numerical test suite including genuinely demanding numerical tests. We observe that the present scheme is able to preserve the physical properties of the numerical solution still ensuring robustness and accuracy when and where appropriate.

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A mass and momentum‐conservative semi‐implicit finite volume scheme for complex non‐hydrostatic free surface flows

Ferrari

Dumbser

2021

Numerical Methods in Fluids

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In this article, a novel mass and momentum conservative semi‐implicit method is presented for the numerical solution of the incompressible free‐surface Navier–Stokes equations. This method can be seen as an extension of the semi‐implicit mass‐conservative scheme presented by Casulli. The domain is covered by the fluid, by potential solid obstacles, and by the surrounding void via a scalar volume fraction function for each phase, according to the so‐called diffuse interface approach. The semi‐implicit finite volume discretization of the mass and momentum equations leads to a mildly nonlinear system for the pressure. The nonlinearity on the diagonal of the system stems from the nonlinear definition of the volume, while the remaining linear part of the pressure system is symmetric and at least positive semi‐definite. Hence, the pressure can be efficiently obtained with the family of nested Newton‐type techniques recently introduced and analyzed by Brugnano and Casulli. The time step size is only limited by the flow speed and eventually by the velocity of moving rigid obstacles contained in the computational domain, and not by the gravity wave speed. Therefore, the method is efficient also for low Froude number flows. Moreover the scheme is formulated to be locally and globally conservative: for this reason it fits well in the presence of shock waves, too. In the special case of only one grid cell in vertical direction, the proposed scheme automatically reduces to a mass and momentum conservative discretization of the shallow water equations. The proposed method is first validated against the exact solution of a set of one‐dimensional Riemann problems for inviscid flows. Then, some computational results are shown for non‐hydrostatic flow problems and for a simple fluid‐structure interaction problem.

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An efficient hyperbolic relaxation system for dispersive non-hydrostatic water waves and its solution with high order discontinuous Galerkin schemes

Cited by 37 publications

References 110 publications

A General Non-hydrostatic Hyperbolic Formulation for Boussinesq Dispersive Shallow Flows and Its Numerical Approximation

A General Non-hydrostatic Hyperbolic Formulation for Boussinesq Dispersive Shallow Flows and Its Numerical Approximation

Solution property preserving reconstruction BVD+MOOD scheme for compressible euler equations with source terms and detonations

A mass and momentum‐conservative semi‐implicit finite volume scheme for complex non‐hydrostatic free surface flows

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