Haut PDF A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes

A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes

A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes

Abstract We present a high-order cell-centered Lagrangian scheme for solving the two-dimensional gas dy- namics equations on unstructured meshes. A node-based discretization of the numerical fluxes for the physical conservation laws allows to derive a scheme that is compatible with the geomet- ric conservation law (GCL). Fluxes are computed using a nodal solver which can be viewed as a two-dimensional extension of an approximate Riemann solver. The first-order scheme is conserva- tive for momentum and total energy, and satisfies a local entropy inequality in its semi-discrete form. The two-dimensional high-order extension is constructed employing the generalized Riemann problem (GRP) in the acoustic approximation. Many numerical tests are presented in order to as- sess this new scheme. The results obtained for various representative configurations of one and two-dimensional compressible fluid flows show the robustness and the accuracy of our new scheme. Key words: Lagrangian hydrodynamics, cell-centered scheme, Generalized Riemann problem, compressible flow, high-order finite volume methods, unstructured mesh
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A high-order cell-centered Lagrangian scheme for compressible fluid flows in two-dimensional cylindrical geometry

A high-order cell-centered Lagrangian scheme for compressible fluid flows in two-dimensional cylindrical geometry

UMR CELIA, Universit´ e Bordeaux I, 351 Cours de la Lib´ eration, 33 405 Talence, France Abstract The goal of this paper is to present high-order cell-centered schemes for solving the equations of Lagrangian gas dynamics written in cylindrical geometry. A node-based discretization of the nu- merical fluxes is obtained through the computation of the time rate of change of the cell volume. It allows to derive finite volume numerical schemes that are compatible with the geometric con- servation law (GCL). Two discretizations of the momentum equations are proposed depending on the form of the discrete gradient operator. The first one corresponds to the control volume scheme while the second one corresponds to the so-called area weighted scheme. Both formulations share the same discretization for the total energy equation. In both schemes, fluxes are computed using the same nodal solver which can be viewed as a two-dimensional extension of an approximate Rie- mann solver. The control volume scheme is conservative for momentum, total energy and satisfies a local entropy inequality in its first-order semi-discrete form. However, it does not preserve spher- ical symmetry. On the other hand, the area weighted scheme is conservative for total energy and preserves spherical symmetry for one-dimensional spherical flow on equiangular polar grid. The two-dimensional high-order extensions of these two schemes are constructed employing the general- ized Riemann problem (GRP) in the acoustic approximation. Many numerical tests are presented in order to assess these new schemes. The results obtained for various representative configurations of one and two-dimensional compressible fluid flows show the robustness and the accuracy of our new schemes.
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A high-order cell-centered finite volume scheme for simulating three dimensional anisotropic diffusion equations on unstructured grids

A high-order cell-centered finite volume scheme for simulating three dimensional anisotropic diffusion equations on unstructured grids

Key words : Finite volume methods, unstructured grids, anisotropic diffusion, parallel computing. 1 Introduction In this paper, we describe a finite volume scheme to solve anisotropic diffusion equations on unstructured grids. This three-dimensional scheme is the natural extension of the two- dimensional scheme CCLAD (Cell-Centered LAgrangian Diffusion) initially presented in [27]. We aim at developing a robust and flexible method for diffusion operators devoted to the numerical modeling of the coupling between heat transfers and fluid flows. More precisely, we are concerned by the numerical simulation of heat transfers in the domain of hypersonic ablation of thermal protection systems [11]. In this context, one has to solve not only the com- pressible Navier-Stokes equations for the fluid flow but also the anisotropic heat equation for the solid materials which compose the thermal protection. These two models, i.e., the Navier- Stokes equations and the heat equation, are strongly coupled by means of a surface ablation model which describes the removal of surface materials resulting from complex thermochem- ical reactions such as sublimation. We point out that in our case, the Navier-Stokes equations
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Positivity-preserving cell-centered Lagrangian schemes for multi-material compressible flows: From first-order to high-orders. Part II: The two-dimensional case

Positivity-preserving cell-centered Lagrangian schemes for multi-material compressible flows: From first-order to high-orders. Part II: The two-dimensional case

traction boundary conditions for all the domain boundaries except the left one which is enforced to be a wall boundary. We recall this problem has neither an analytical solution nor experimental results. However, it is a good test to assess the robustness of our numerical method, while keeping in mind that the numerical solution accuracy will be limited by equations model studied, which is in our case the compressible gas dynamics system. In Figure 15, we have displayed the initial grid and the final density maps obtained at time t = 0.005 by means of the first-order and limited second- order DG schemes, with the acoustic solver, on a 100 × 10 Cartesian grid. Comparing Figures 15(b) and 15(c), we clearly see the second-order final solution grid a lot more deformed than the first-order one. This phenomenon can be explained by the large amount of numerical diffusion inherent of the first-order scheme. As it has been demonstrated in [43] in the case of a Gresho-like vortex problem, in some extreme cases the first-order scheme is unable to simulate appropriately the problems to the final time due to the large numerical diffusion. But obviously, resolving the compressible gas dynamics equations, the numerical schemes presented are not able to capture elastic waves and features. This is the reason why the second-order results, Figure 15(c), are quite far from the ones obtained in [31] with an elastic-plastic second-order cell-centered Lagrangian scheme. Nevertheless, this test case permitted us once more to prove the robustness of the cell-centered Lagrangian schemes presented, as no negative density or negative internal energy appears during the calculation. 13. Conclusion
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Positivity-preserving cell-centered Lagrangian schemes for multi-material compressible flows: From first-order to high-orders. Part I: The one-dimensional case

Positivity-preserving cell-centered Lagrangian schemes for multi-material compressible flows: From first-order to high-orders. Part I: The one-dimensional case

of neighboring cells sharing a node, one cannot apply in a straightforward manner one-dimensional solvers to define uniquely the grid point velocity. The staggered hydrodynamics has been developed to avoid such complications. In this framework, a staggered discretization is employed such that the kinematic variables (vertex position, velocity) are located at the nodes whereas the thermody- namic variables (density, pressure, internal energy) are defined at the cell centers. The conversion of kinetic energy into internal energy through shock waves, consistent with the second law of thermo- dynamics, is ensured by adding an artificial viscosity term. The staggered grid schemes employed in most hydro-codes have been remarkably successful over the past decades in solving complex multi-dimensional compressible fluid flows, refer for instance to [45, 88, 17, 18, 32, 55, 33, 34, 4]. However, they clearly have some theoretical and practical deficiencies such as mesh imprinting and symmetry breaking. In addition, the fact that all variables are not conserved over the same space can make these schemes difficult to handle when one wants to assess analytical properties of the numerical solution. For all these reasons, this paper focuses on the cell-centered approach. Different techniques may be employed to build the numerical fluxes and move the grid through the use of approximate Riemann solvers, with respect to the GCL. The interested readers may refer to the following papers [1, 19, 20, 61, 72, 15, 53, 82, 5, 16, 10, 84] for a more detailed description of this approach and its variants. Let us mention that besides these two mainly employed approaches, i.e. the cell-centered and staggered approaches, a third one has recently grows quickly in popular- ity these past years. This third framework, referred to as Point-Centered Hydrodynamic (PCH), combines the features of the first two, namely a dual grid and the fact that kinetical and thermody- namical variables are conserved on the same cells. Indeed, in this particular frame, the momentum and total energy conservation equations are solved on the dual grid around the nodes, generally by means of an edge-based finite element scheme or an edge-based upwind finite volume method. The PCH approach has been successful applied these past decades to problematics concerned with the simulation of incompressible flows, compressible Lagrangian flows, or Lagrangian solid dynamics, refer for instance to [28, 29, 24, 42, 48, 76, 79, 78, 86, 87, 67, 68, 2, 3]. Two of the main advantages of these schemes are that there are very well adapted to triangular or tetrahedral grids, as well as they reduce in most cases problems related to mesh stiffness.
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Positivity-preserving cell-centered Lagrangian schemes for multi-material compressible flows: From first-order to high-orders

Positivity-preserving cell-centered Lagrangian schemes for multi-material compressible flows: From first-order to high-orders

one-dimensional solvers to define uniquely the grid point velocity. The staggered hydrodynamics has been developed to avoid such complications. In this framework, a staggered discretization is employed such that the kinematic variables (vertex position, velocity) are located at the nodes whereas the thermodynamic variables (density, pressure, internal energy) are defined at the cell centers. The conversion of kinetic energy into internal energy through shock waves, consistent with the second law of thermodynamics, is ensured by adding an artificial viscosity term. The staggered grid schemes employed in most hydro-codes have been remarkably successful over the past decades in solving complex multi-dimensional compressible fluid flows, refer for instance to [33, 73, 10, 11, 45]. However, they clearly have some theoretical and practical deficiencies such as mesh imprinting and symmetry breaking. In addition, the fact that all variables are not conserved over the same space can make these schemes difficult to handle when one wants to assess analytical properties of the numerical solution. For all these reasons, this paper focuses on the cell-centered approach. Different techniques may be employed to build the numerical fluxes and move the grid through the use of approximate Riemann solvers, with respect to the GCL. The interested readers may refer to the following papers [1, 12, 13, 52, 63, 8, 41, 70, 3, 9, 7, 72] for a more detailed de- scription of this approach and its variants. In the section dedicated to the two-dimensional case, a general procedure to develop first-order finite volume schemes on general polygonal meshes is presented. Such formulation will cover the numerical methods introduced in [12, 55, 8, 70, 72]. Although a wide range of purely Lagrangian formulations are available, together with the different advantages associated to them, it is well known these descriptions admit a severe drawback in some situations. In the presence of intense vortexes or
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A two-dimensional unstructured cell-centered multi-material ALE scheme using VOF interface reconstruction

A two-dimensional unstructured cell-centered multi-material ALE scheme using VOF interface reconstruction

This paper aims at presenting a cell-centered indirect ALE algorithm to solve multi-material compressible flows on two-dimensional unstructured grids with fixed topology. Our Lagrangian phase solves the gaz dynamics equations utilizing a moving mesh cell-centered discretization wherein the physical conservation laws are discretized in a compatible manner with the nodal velocity so that the geometric conservation law (GCL) is exactly satisfied [12]. Namely, the time rate of change of a Lagrangian volume is computed consistently with the node motion. This critical requirement is the cornerstone of any Lagrangian multidimensional scheme. Nowadays, cell-centered finite volume schemes [11, 36, 35] that fulfill this GCL requirement seem to be a promising alternative to the usual staggered finite difference discretization [10]. Moreover, these cell-centered schemes allow straightforward implementation of conservative remapping methods when they are used in the context of ALE. Here, we are using the high-order cell-centered Lagrangian scheme that has been described in [35]. Let us recall that the numerical fluxes are determined by means of a node-centered approximate Riemann solver. This discretization leads to a conservative and entropy consistent scheme whose high-order extension is derived through the use of generalized Riemann problem [7, 35].
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A Cell-Centered Lagrangian Scheme for Two-Dimensional Compressible Flow Problems

A Cell-Centered Lagrangian Scheme for Two-Dimensional Compressible Flow Problems

is an accurate and robust method, which can produce impressive results, even on unstructured polygonal grids, see for instance [21]. An alternative to the staggered discretization is to use a conservative cell-centered discretization. This method for Lagrangian gas dynamics in one dimension, has been introduced by Godunov, see [12] and [24]. The multidimensional extension of this method has been performed during the eighties, [2] and [11]. This multidimensional scheme is a cell-centered finite volume scheme on moving structured or unstructured meshes. It is constructed by integrating directly the system of conservation laws on each moving cell. The primary variables, density, momentum and total energy are defined in the cells. The flux across the boundary of the cell is computed by solving exactly or approximately a one-dimensional Riemann problem in the direction normal to the boundary. The main problem with this type of method lies in the fact that the node velocity needed to move the mesh cannot be directly calculated. In [2], the node velocity is computed via a special least squares procedure. It consists in minimizing the error between the normal velocity coming from the Riemann solver and the normal projection of the vertex velocity. It turns out that it leads to an artificial grid motion, which requires a very expensive treatment [10]. Moreover, with this approach the flux calculation is not consistent with the node motion.
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A high-order finite volume cell-centered scheme for anisotropic diffusion on two-dimensional unstructured grids

A high-order finite volume cell-centered scheme for anisotropic diffusion on two-dimensional unstructured grids

 sin θ, where (r, θ) are the polar coordinates corresponding to the Cartesian coordinates (x, y). The corresponding distorted grid obtained setting a0 = 0.25 and n = 8 is plotted in Figure 22(b). Now, we compute the numerical solution of the non-linear test problem on the above distorted grid, using two different schemes. These are: the classical five-point scheme and the CCLADS scheme which reduces to a nine-point scheme on quadrangular grids. The numerical solution resulting from the five-point scheme is plotted in Figure 23 using blue dots. We have displayed the temperatures in all cells as function of the cell center radius versus the reference solution. In this figure, we observe the main flaw of the five-point scheme: in spite of its robustness, it produces a numerical solution wherein the temperature front is aligned with the grid distortion. The corresponding numerical solution is not able to preserve the cylindrical symmetry. In addition, the comparison to the reference solution, shows that the timing of the thermal wave is completely wrong. Let us emphasize that this test case is not a fake problem. It is representative of situations which frequently occur in the framework of plasma physics simulation wherein the heat conduction equation is coupled with a numerical method solving Lagrangian hydrodynamics equations. In this case, grid distortions are induced by fluid motion and thus the use of the five-point scheme to solve the heat conduction equation leads to a very bad result. This weakness of the five-point scheme follows from the fact that its construction is based on a two-point flux approximation 3 , which becomes inaccurate in the
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A nominally second-order cell-centered Lagrangian scheme for simulating elastic-plastic flows on two-dimensional unstructured grids

A nominally second-order cell-centered Lagrangian scheme for simulating elastic-plastic flows on two-dimensional unstructured grids

non structur´ es bidimensionnels R´ esum´ e : In this paper, we describe a cell-centered Lagrangian scheme devoted to the numeri- cal simulation of solid dynamics on two-dimensional unstructured grids in planar geometry. This numerical method, utilizes the classical elastic-perfectly plastic material model initially proposed by Wilkins [M.L. Wilkins, Calculation of elastic-plastic flow, Meth. Comput. Phys. (1964)]. In this model, the Cauchy stress tensor is decomposed into the sum of its deviatoric part and the thermodynamic pressure which is defined by means of an equation of state. Regarding the devi- atoric stress, its time evolution is governed by a classical constitutive law for isotropic material. The plasticity model employs the von Mises yield criterion and is implemented by means of the radial return algorithm. The numerical scheme relies on a finite volume cell-centered method wherein numerical fluxes are expressed in terms of sub-cell force. The generic form of the sub-cell force is obtained by requiring the scheme to satisfy a semi-discrete dissipation inequality. Sub-cell force and nodal velocity to move the grid are computed consistently with cell volume variation by means of a node-centered solver, which results from total energy conservation. The nominally second-order extension is achieved by developing a two-dimensional extension in the Lagrangian framework of the Generalized Riemann Problem methodology, introduced by Ben-Artzi and Fal- covitz [M. Ben-Artzi and J. Falcovitz, Generalized Riemann Problems in Computational Fluid Dynamics, Cambridge Monographs on Applied and Computational Mathematics, 2003]. Finally, the robustness and the accuracy of the numerical scheme are assessed through the computation of several test cases.
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A High-Order Monotonicity-Preserving Scheme for Linear Scalar Advection on 3-D Irregular Meshes

A High-Order Monotonicity-Preserving Scheme for Linear Scalar Advection on 3-D Irregular Meshes

Figure 7 depicts the error distribution in the plane z = 0.009 of the display cube, for the conservative transport of a SQUARE. In the ISE S panel, there is no conservativity-correction applied at each time iteration, while in the ISE Z panel, the conservativity-correction procedure is activated. We clearly see that the error is much bigger for the DON scheme. To be convinced of this fact, let us proceed to 1-D cuts along the 3 principal directions of the display cube. In Fig. 8, we compare the 3 schemes in each cut panel. As far as the amplitude is concerned, ISE S and ISE Z are quite similar. As for DON, it is obviously worse by one order of magnitude. Figures 9 and 10 plot the same results but associated with the WAVELET data. The observa- tions are even more impressive here, insofar as no scheme is good enough to predict to maximal value, but DON remains the most diffusive. One can wonder why the situation is worse for WAVELET, a smooth function, than for SQUARE, a discontinuous data. The answer is: although we are working with highly accurate schemes, we are still very far from convergence, because the (deformed) mesh we are using is very coarse with respect to the data.
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A constrained transport scheme for MHD on unstructured static and moving meshes

A constrained transport scheme for MHD on unstructured static and moving meshes

100 kpc-scale high-density arms to exist in the cluster environment (Sanders et al. 2013). The strength and topology of magnetic fields are responsible for determining the propagation of cosmic rays in galaxies (Strong & Moskalenko 1998). Magnetic fields are also present in a wide variety of stars and play a significant role in their evolution (Donati & Landstreet 2009). Many of these problems are suitable for study by a moving mesh approach and up until now a divergence-free MHD solver has been lacking. This paper lays the framework for implementing a divergence-preserving algorithm for robust evolution of the MHD equations in moving mesh codes such as AREPO (Springel 2010) and TESS (Duffell & MacFadyen 2011). In Section 2 we describe the details of the numerical method. In Section 3 we show the results of numerical tests (with compar- isons to fixed grid CT and the Powell cleaning scheme on a moving mesh) demonstrating that the method works well and has several advantages over the other two techniques. In Section 4 we discuss variations of our method and future directions. In Section 5 we pro- vide concluding remarks.
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A first-order bi-projection scheme for incompressible two-phase Bingham flows

A first-order bi-projection scheme for incompressible two-phase Bingham flows

on a staggered mesh. The staggered location of the velocity unknowns allows to compute a free-divergence velocity field, up to the computer accuracy, which is used as transport velocity in the mass conservation equation. Numerical simulations of Rayleigh-Taylor instabilities are performed and presented. The first order rate of convergence with respect to the time step is recovered both in the case of Newtonian and Bingham flows. Comparisons with published results in the case of Newtonian flows validate the parallel implementation of the bi-projection scheme. Numerical simulations of a Bingham flow at Bi = 10 and 100 are reported and discussed. From the best of our knowledge, these results are the first ones obtained for this problem. At even larger Bingham number, we may expect that the heavier fluid will not flow. Investigating in more details the dynamics and behavior of Rayleigh-Taylor instabilities when the Bingham number is increased could be the scope of further works.
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Sharpening diffuse interfaces with compressible fluids on unstructured meshes

Sharpening diffuse interfaces with compressible fluids on unstructured meshes

Diffuse interface methods with compressible fluids, considered through hyperbolic multiphase flow models, have demonstrated their capability to solve a wide range of complex flow situations in severe conditions (both high and low speeds). These formulations can deal with the pres- ence of shock waves, chemical and physical transformations, such as cavitation and detonation. Compared to existing approaches able to consider compressible materials and interfaces, these methods are conservative with respect to mixture mass, momentum, energy and are entropy preserving. Thanks to these properties they are very robust. However, in many situations, typ- ically in low transient conditions, numerical diffusion at material interfaces is excessive. Several approaches have been developed to lower this weakness. In the present contribution, a specific flux limiter is proposed and inserted into conventional MUSCL type schemes, in the frame of the diffuse interface formulation of Saurel et al. (2009). With this limiter, interfaces are captured with almost two mesh points at any time, showing significant improvement in interface represen- tation. The method works on both structured and unstructured meshes and its implementation in existing codes is simple. Computational examples showing method capabilities and accuracy are presented.
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Low-Mach correction for Lagrangian acoustic Riemann solvers on unstructured meshes

Low-Mach correction for Lagrangian acoustic Riemann solvers on unstructured meshes

velocity’s unknowns are associated with the vertices of the cells, while the ther- modynamics’s unknowns (internal energy, density) are associated with the cells of the mesh. This is a natural choice for semi-Lagrangian methods, because it directly provides a procedure to move the mesh compatibly with the velocity unknowns. This intrinsic quality justifies the great number of works devoted to this method for more than 60 years. Among these works, some concern the design of artificial viscosity. Artificial viscosity is needed in staggered schemes, because the discretization of the pressure gradient is by construction "centered" and does not account for the entropy increase in shock waves. The artificial viscosity cures this flaw in adding a dissipative term proportional to the ve- locity jump into the momentum equation. Modern versions of this mechanism include a monotonic second-order reconstruction of this jump, in order to dis- card the artificial viscosity for regular (isentropic) flows. One indirect and never mentioned consequence of this enhancement is to widely improve the behavior of staggered schemes in low-Mach regimes. Indeed, it has been shown that a correct calculation of low-Mach flows is directly related to a centered pressure gradient (refer for instance to [ 19 ]).
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Simulation of Free Surface Compressible Flows Via a Two Fluid Model

Simulation of Free Surface Compressible Flows Via a Two Fluid Model

the maximum pressure and impact duration differed from one identical wave impact to the next, even in carefully controlled laboratory experiments, while the pressure impulse appears to be more repeatable. For sure, the simple one-fluid models which are commonly used for examining the peak impacts are no longer appropriate in the presence of air. There are few studies dealing with two-fluid models. An exception is the work by Peregine and his collaborators. Wood, Peregrine & Bruce [4] used the pres- sure impulse approach to model a trapped air pocket. Peregrine & Thais [5] examined the effect of entrained air on a particu- lar kind of violent water wave impact by considering a filling flow. Bullock et al. [6] found pressure reductions when compar- ing wave impact between fresh and salt water where, due to the different properties of the bubbles in the two fluids, the aeration levels are much higher in salt water than in fresh. H. Bredmose recently performed numerical experiments on a two-fluid system which is quite similar to the one we will use below. He developed a finite volume solver for aerated flows named Flair [7].
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Simulation of compressible viscous flows on a variety pf MPPs : computational algorithms for unstructured dynamic meshes and performance results

Simulation of compressible viscous flows on a variety pf MPPs : computational algorithms for unstructured dynamic meshes and performance results

Unite´ de recherche INRIA Lorraine, Technoˆpole de Nancy-Brabois, Campus scientifique, 615 rue de Jardin Botanique, BP 101, 54600 VILLERS LE`S NANCY Unite´ de recherche INRIA Rennes, IRI[r]

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A second-order realizable scheme for moment advection on unstructured grids

A second-order realizable scheme for moment advection on unstructured grids

The nature of the 𝜁 scheme and, in particular, the version of this scheme called 𝜁 simplified scheme by Laurent and Nguyen [1], makes it an ideal candidate for its extension to unstructured grids with cells of arbitrary shapes because the scheme relies on a traditional MUSCL reconstruction, and only requires a local additional limitation to be applied to the reconstructed quantities on cell faces. The extension of this scheme to unstructured grids and its implementation into the OpenFOAM framework [56] are the topics of this article, the remainder of which is organized as follows: in Sec. 2 the problem of moment transport is introduced. The 𝜁 simplified scheme for hexahedral structured grids of Laurent and Nguyen [1] is summarized in Sec. 3. Its generalization to unstructured grids with cells of arbitrary shapes is discussed in Sec. 4. Finally, the same one- and two-dimensional test cases used by Laurent and Nguyen [1] are used in Sec. 5 to verify the implementation of the numerical scheme, using hexahedral uniform grids. Then, the two-dimensional cases are repeated using a triangular grid, and the order of accuracy of the 𝜁 simplified scheme on unstructured grids is assessed.
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Block-based adaptive mesh refinement scheme using numerical density of
entropy production for three-dimensional two-fluid flows

Block-based adaptive mesh refinement scheme using numerical density of entropy production for three-dimensional two-fluid flows

The model is based on an artificial linearized pressure law for low Mach regime. This artificial pressure law induces several interesting effects from mathematical and numerical viewpoints described above and in previous related papers. The novelty of the present work is to extend the multi-scale adaptive scheme proposed by [10] to our three-dimensional hyperbolic equations. The automatic mesh refinement process is performed thanks to an original criterion: the numerical density of entropy production. The increase of computational costs caused by the re-meshing procedure is outweighed by the introduction of an efficient Block-Based Adaptive Mesh Refinement method (BB-AMR) allowing an easy and fast parallel implementa- tion and computation. Through several 2D and 3D wave impact test cases, the overall numerical approach is validated. In particular, the entropy production appears to be a relevant criterion for automatic mesh refinement process, which leads to better describe the regions of interest (interfaces and impact zones) and to drastically reduce computational time.
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Construction of very high order residual distribution schemes for steady inviscid flow problems on hybrid unstructured meshes

Construction of very high order residual distribution schemes for steady inviscid flow problems on hybrid unstructured meshes

along the boundary . Ex ept at the forefront stagnation point, the entropy de- viation of the third order s heme is mu h loser than the exa t one. We have re-run this test ase on an hybrid mesh using the se ond order and the third order s hemes. In both ases, the same degrees of freedom are used

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