Well-balanced Arbitrary-Lagrangian-Eulerian finite volume schemes on moving nonconforming meshes for the Euler equations of gas dynamics with gravity

Gaburro, Elena; Castro, Manuel J.; Dumbser, Michael

doi:10.1093/mnras/sty542

Cited by 54 publications

(49 citation statements)

References 69 publications

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“…We solve the two-phase model (16) with a second order well balanced path-conservative scheme based on the Osher-Romberg numerical flux introduced for the first time for the shallow water equations in cylindrical coordinates in [25] and for the Euler equations with gravity in [36].…”

Section: Well Balanced Path-conservative Schemementioning

confidence: 99%

“…The concept of a path-conservative method, which is also based on a prescribed family of paths, provides a generalization of conservative schemes introduced by Lax for systems of conservation laws. Then we will apply the recently introduced Osher-Romberg numerical flux [25,36] which is a general and very accurate well balanced technique. Provided the full set of eigenvalues and eigenvectors of A are known, it can be easily adapted to very different systems of equations and families of equilibrium solutions.…”

Section: Introductionmentioning

confidence: 99%

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A well balanced diffuse interface method for complex nonhydrostatic free surface flows

2018

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In this paper we propose an efficient second order accurate well balanced finite volume method for modeling complex free surface flows at the aid of a simple diffuse interface method. The employed physical model is a two-phase model directly derived from the Baer-Nunziato system for compressible multi-phase flows. In particular, as proposed for the first time in [1], the number of equations is reduced from seven to three by assuming that the relative pressure of the gas with respect to the atmospheric reference pressure is zero, and that the gas momentum is negligible compared to the one of the liquid. The two-phase model does not make any of the classical assumptions of shallow water type systems, hence it does not neglect vertical accelerations and the free surface is not constraint to be a single-valued function, so even complex shapes as those of breaking waves can be properly captured.The resulting PDE system is solved by a novel well balanced second order accurate path-conservative finite volume method on structured Cartesian grids, which is able to preserve exactly the equilibrium states even in the presence of obstacles. It furthermore automatically computes the location of the water-air interfaces, and assures low numerical dissipation at the free surface thanks to a novel Osher-Romberg-type approximate Riemann solver. Finally, high computational performance is guaranteed by an efficient parallel implementation on a GPU-based platform that reaches the efficiency of twenty million of volumes processed per seconds and makes it possible to employ even very fine meshes. The validation of our new well balanced scheme is carried out by comparing the obtained numerical results against existing analytical, numerical and experimental reference solutions for a large number of test cases, among which oscillating elliptical drops, dambreak problems, breaking waves, over topping weir flows, and wave impact problems.Key words: diffuse interface method, reduced Baer-Nunziato model of compressible multi-phase flows, well balanced path-conservative method, Osher-Romberg flux, parallel GPU implementation based on NVIDIA CUDA, complex free surface flow.

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Section: Well Balanced Path-conservative Schemementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A well balanced diffuse interface method for complex nonhydrostatic free surface flows

2018

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“…In the last decades, the scientific community has put a lot of efforts into the study of these phenomena, see e.g. [5,6,7,8,9,10] for a non-exhaustive overview. Nowadays, the main challenge is to develop efficient high order numerical methods which are able to capture even small scale structures of the flow, avoiding the use of RANS turbulence models (see [11,12]).…”

Section: Introductionmentioning

confidence: 99%

Efficient high order accurate staggered semi-implicit discontinuous Galerkin methods for natural convection problems

et al. 2020

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In this article we propose a new family of high order staggered semi-implicit discontinuous Galerkin (DG) methods for the simulation of natural convection problems. Assuming small temperature fluctuations, the Boussinesq approximation is valid and in this case the flow can simply be modeled by the incompressible Navier-Stokes equations coupled with a transport equation for the temperature and a buoyancy source term in the momentum equation. Our numerical scheme is developed starting from the work presented in [1,2,3], in which the spatial domain is discretized using a face-based staggered unstructured mesh. The pressure and temperature variables are defined on the primal simplex elements, while the velocity is assigned to the dual grid. For the computation of the advection and diffusion terms, two different algorithms are presented: i) a purely Eulerian explicit upwind-type scheme and ii) a semi-Lagrangian approach. The first methodology leads to a conservative scheme whose major drawback is the time step restriction imposed by the CFL stability condition due to the explicit discretization of the convective terms. On the contrary, computational efficiency can be notably improved relying on a semi-Lagrangian approach in which the Lagrangian trajectories of the flow are tracked back. This method leads to an unconditionally stable scheme if the diffusive terms are discretized implicitly. Once the advection and diffusion contributions have been computed, the pressure Poisson equation is solved and the velocity is updated. As a second model for the computation of buoyancy-driven flows, in this paper we also consider the full compressible Navier-Stokes equations. The staggered semi-implicit DG method first proposed in [4] for all Mach number flows is properly extended to account for the gravity source terms arising in the momentum and energy conservation laws. In order to assess the validity and the robustness of our novel class of staggered semi-implicit DG schemes, several classical benchmark problems are considered, showing in all cases a good agreement with available numerical reference data. Furthermore, a detailed comparison between the incompressible and the compressible solver is presented. Finally, advantages and disadvantages of the Eulerian and the semi-Lagrangian methods for the discretization of the nonlinear convective terms are carefully studied.

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“…The rest of this article is structured as follows: in Section 2, we detail the derivation of our diffuse interface model describing compressible flows around moving solid obstacles, and in particular we provide a detailed proof that the solid and the gas velocities are equal at the material interface; then in Section 3, we briefly describe the numerical scheme employed for our simulations based on high order ADER-DG schemes with a posteriori sub-cell finite volume limiter for dealing with discontinuities. The non-conservative products are treated via the path-conservative approach of Parés and Castro [16,103,17,60]. The obtained numerical results are shown in Section 4, and finally, in Section 5, we give some concluding remarks and an outlook to future research and developments.…”

Section: Introductionmentioning

confidence: 99%

A simple diffuse interface approach for compressible flows around moving solids of arbitrary shape based on a reduced Baer–Nunziato model

et al. 2020

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In this paper we propose a new diffuse interface model for the numerical simulation of inviscid compressible flows around fixed and moving solid bodies of arbitrary shape. The solids are assumed to be moving rigid bodies, without any elastic properties. The mathematical model is a simplified case of the seven-equation Baer-Nunziato model of compressible multi-phase flows. The resulting governing PDE system is a nonlinear system of hyperbolic conservation laws with non-conservative products. The geometry of the solid bodies is simply specified via a scalar field that represents the volume fraction of the fluid present in each control volume. This allows the discretization of arbitrarily complex geometries on simple uniform or adaptive Cartesian meshes. Inside the solid bodies, the fluid volume fraction is zero, while it is unitary inside the fluid phase. Due to the diffuse interface nature of the model, the volume fraction function can assume any value between zero and one in mixed cells that are occupied by both, fluid and solid.We also prove that at the material interface, i.e. where the volume fraction jumps from unity to zero, the normal component of the fluid velocity assumes the value of the normal component of the solid velocity. This result can be directly derived from the governing equations, either via Riemann invariants or from the generalized Rankine Hugoniot conditions according to the theory of Dal Maso, Le Floch and Murat [89], which justifies the use of a path-conservative approach for treating the non-conservative products.The governing partial differential equations of our new model are solved on simple uniform Cartesian grids via a high order path-conservative ADER discontinuous Galerkin (DG) finite element method with a posteriori sub-cell finite volume (FV) limiter. Since the numerical method is of the shock capturing type, the fluid-solid boundary is never explicitly tracked by the numerical method, neither via interface reconstruction, nor via mesh motion.The effectiveness of the proposed approach is tested on a set of different numerical test problems, including 1D Riemann problems as well as supersonic flows over fixed and moving rigid bodies.Key words: diffuse interface model, compressible flows over fixed and moving solids, immersed boundary method for compressible flows, arbitrary high-order discontinuous Galerkin schemes, a posteriori sub-cell finite volume limiter (MOOD), path-conservative schemes for hyperbolic PDE with non-conservative products,

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Well-balanced Arbitrary-Lagrangian-Eulerian finite volume schemes on moving nonconforming meshes for the Euler equations of gas dynamics with gravity

Cited by 54 publications

References 69 publications

A well balanced diffuse interface method for complex nonhydrostatic free surface flows

A well balanced diffuse interface method for complex nonhydrostatic free surface flows

Efficient high order accurate staggered semi-implicit discontinuous Galerkin methods for natural convection problems

A simple diffuse interface approach for compressible flows around moving solids of arbitrary shape based on a reduced Baer–Nunziato model

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