A well-balanced path conservative SPH scheme for nonconservative hyperbolic systems with applications to shallow water and multi-phase flows

Rossi, Giulia; Dumbser, Michael; Armanini, Aronne

doi:10.1016/j.compfluid.2017.05.034

Cited by 15 publications

(6 citation statements)

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“…All that is needed is to replace the conventional algebraic source term by our new well balanced path-conservative approximate Riemann solver, which interprets the gravity source term as a nonconservative product. Nevertheless, using the novel ideas on well balanced SPH methods very recently presented in Rossi et al (2017), it seems also possible to extend the new well balanced approach for the Euler equations with gravity presented here to Smooth Particle Hydrodynamics. However, this is beyond the scope of the present paper and its feasibility will be subject to further investigations.…”

Section: Discussionmentioning

confidence: 99%

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

Gaburro

Castro

Dumbser

2018

Monthly Notices of the Royal Astronomical Society

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In this work we present a novel second order accurate well balanced Arbitrary-Lagrangian-Eulerian (ALE) finite volume scheme on moving nonconforming meshes for the Euler equations of compressible gasdynamics with gravity in cylindrical coordinates. The main feature of the proposed algorithm is the capability of preserving many of the physical properties of the system exactly also on the discrete level: besides being conservative for mass, momentum and total energy, also any known steady equilibrium between pressure gradient, centrifugal force and gravity force can be exactly maintained up to machine precision. Perturbations around such equilibrium solutions are resolved with high accuracy and with minimal dissipation on moving contact discontinuities even for very long computational times. This is achieved by the novel combination of well balanced path-conservative finite volume schemes, that are expressly designed to deal with source terms written via nonconservative products, with ALE schemes on moving grids, which exhibit only very little numerical dissipation on moving contact waves. In particular, we have formulated a new HLL-type and a novel Osher-type flux that are both able to guarantee the well balancing in a gas cloud rotating around a central object. Moreover, to maintain a high level of quality of the moving mesh, we have adopted a nonconforming treatment of the sliding interfaces that appear due to the differential rotation. A large set of numerical tests has been carried out in order to check the accuracy of the method close and far away from the equilibrium, both, in one and two space dimensions.

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Section: Discussionmentioning

confidence: 99%

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

Gaburro

Castro

Dumbser

2018

Monthly Notices of the Royal Astronomical Society

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“…These test cases have been widely used by other researchers to test different mesh-based numerical methods, in 1D [6,49,54] and in 2D [6,12,54,55]. For SPH methods, these test cases have been used in [29,31]. A C F L = 0.15 was employed in these cases.…”

Section: Propagation Of Small Water Surface Perturbations On Water At Rest Casesmentioning

confidence: 99%

“…We can refer to previous works from Vacondio et al [27,28], Xia et al [29] and Berthon et al [30] . To the authors' knowledge, the work of Rossi et al [31] is the only successful attempt employing the SPH-ALE formulation. In their work, Rossi et al presented a path-conservative scheme.…”

Section: Introductionmentioning

confidence: 99%

A Well-Balanced SPH-ALE Scheme for Shallow Water Applications

et al. 2021

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In this work, a new discretization of the source term of the shallow water equations with non-flat bottom geometry is proposed to obtain a well-balanced scheme. A Smoothed Particle Hydrodynamics Arbitrary Lagrangian-Eulerian formulation based on Riemann solvers is presented to solve the SWE. Moving-Least Squares approximations are used to compute high-order reconstructions of the numerical fluxes and, stability is achieved using the a posteriori MOOD paradigm. Several benchmark 1D and 2D numerical problems are considered to test and validate the properties and behavior of the presented schemes.

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“…The benchmark test proposed by LeVeque (1998) has been widely used to evaluate the well‐balancing property of numerical schemes to simulate rapidly varied flow over a non‐flat bottom topography (Heng et al., 2009; Rossi et al., 2017; Xia et al., 2013). In a well‐balanced scheme, no discontinuity is observed neighboring the non‐flat topography.…”

Section: Model Verification and Validationmentioning

confidence: 99%

“…utilized an upwind scheme employing a tuning parameter to determine the upwinding degree for SPH discretization instead of using an artificial viscosity. Rossi et al (2017) stated that approximate Riemann Solvers, namely Rusanov, Osher-type DOT, and HL-LEM methods, have intrinsic artificial viscosity that can be included in the SPH schemes of SWF models instead of the traditional artificial viscosity. They also conducted a series of comparisons between various approximate Riemann Solvers with multiple frictionless flow cases and documented the new stabilization technique's efficiency.…”

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confidence: 99%

MPS‐Based Model to Solve One‐Dimensional Shallow Water Equations

Sarkhosh

Jin

2021

Water Resources Research

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Two meshless (or mesh-free) methods that are widely employed in hydrodynamic modeling of problems with steep gradients and large deformation are smoothed particle hydrodynamics (SPH) and moving particle simulation (MPS) that are classified into two categories of incompressible and weakly compressible fluids (Gotoh & Khayyer, 2016). Although SPH was initially introduced for astrophysical applications (Gingold & Monaghan, 1977), in the scope of hydrodynamics, SPH and MPS were developed by Monaghan (1992) and Koshizuka and Oka (1996), respectively, for solving the free-surface incompressible flow using Navier-Stokes equations. Gotoh et al. ( 2001) presented a sub-particle scale closure model for closing large eddy simulation (LES) to simulate turbulence fluctuations in the MPS method. The theory of a weakly compressible flow was adopted by Monaghan (1994) in SPH and Shakibaeinia and Jin (2010) in MPS using a thermodynamic equation of state for calculating the pressure instead of resorting to the Poisson equation.The main difference between SPH and MPS arises from different differential operators employed in the spatial integration of the partial differential equations. In SPH, the differential operators are obtained from the differentiation of a weighted average function, the so-called kernel, without directly applying differencing schemes for flow variables. The superposition of the kernels then computes flow variables' derivatives (Liu & Liu, 2010). In contrast, MPS is the extension of the finite volume method to a mesh-free approach in which the spatial discretization is obtained based on the differentiation of flow variables. The weighted average of physical quantities derivatives of neighboring particles is applied to the target particle. However, unlike the finite volume method in which the velocity divergence is used for pressure calculation, the particle number density is adopted in MPS. The particle number density indicating the number of particles surrounding a particle is a key variable in MPS that is straightforward to fulfill flow incompressibility (Koshizuka et al., 2018).

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A well-balanced path conservative SPH scheme for nonconservative hyperbolic systems with applications to shallow water and multi-phase flows

Cited by 15 publications

References 57 publications

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

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

A Well-Balanced SPH-ALE Scheme for Shallow Water Applications

MPS‐Based Model to Solve One‐Dimensional Shallow Water Equations

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