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We present a novel Finite Volume (FV) scheme on unstructured polygonal meshes that is provably compliant with the Second Law of Thermodynamics and the Geometric Conservation Law (GCL) at the same time. The governing equations are provided by a subset of the class of symmetric and hyperbolic thermodynamically compatible (SHTC) models introduced by Godunov in 1961. Specifically, our numerical method discretizes the equations for the conservation of momentum, total energy, distortion tensor and thermal impulse vector, hence accounting in one single unified mathematical formalism for a wide range of physical phenomena in continuum mechanics, spanning from ideal and viscous fluids to hyperelastic solids. By means of two conservative corrections directly embedded in the definition of the numerical fluxes, the new schemes are proven to satisfy two extra conservation laws, namely an entropy balance law and a geometric equation that links the distortion tensor to the density evolution. As such, the classical mass conservation equation can be discarded. Firstly, the GCL is derived at the continuous level, and subsequently it is satisfied by introducing the new concepts of general potential and generalized Gibbs relation. The new potential is nothing but the determinant of the distortion tensor, and the associated Gibbs relation is derived by introducing a set of dual or thermodynamic variables such that the GCL is retrieved by dot multiplying the original system with the new dual variables. Once compatibility of the GCL is ensured, thermodynamic compatibility is tackled in the same manner, thus achieving the satisfaction of a local cell entropy inequality. The two corrections are orthogonal, meaning that they can coexist simultaneously without interfering with each other. The compatibility of the new FV schemes holds true at the semi-discrete level, and time integration of the governing PDE is carried out relying on Runge-Kutta schemes. A large suite of test cases demonstrates the structure preserving properties of the schemes at the discrete level as well. • Thermodynamic compatibility. • Geometric compatibility on fixed grids. • Overdetermined systems with extra conservation laws. • Finite Volume schemes on general unstructured meshes. • Unified model of continuum mechanics for fluids and solids. [ABSTRACT FROM AUTHOR]

In this paper we blend the high order Compact Approximate Taylor (CAT) numerical schemes with an a posteriori Multi-dimensional Optimal Order Detection (MOOD) paradigm to solve hyperbolic systems of conservation laws in 2D. The resulting scheme presents high accuracy on smooth solutions, essentially non-oscillatory behavior on irregular ones, and almost fail-safe property concerning positivity issues. The numerical results on a set of sanity test cases and demanding ones are presented assessing the appropriate behavior of the CAT-MOOD scheme. • Hig-order one-step Lax-Wendroff type scheme for 2D systems of conservation laws. • Positivity-preserving and conservative scheme. • Finite-difference scheme. • A-posteriori order adaptive scheme. • 2D numerical experiments as Euler equations with smooth and non-smooth initial conditions. [ABSTRACT FROM AUTHOR]

In this paper we propose a third order accurate finite volume scheme based on a posteriori limiting of polynomial reconstructions within an Adaptive-Mesh-Refinement (AMR) simulation code for hydrodynamics equations in 2D. The a posteriori limiting is based on the detection of problematic cells on a so-called candidate solution computed at each stage of a third order Runge–Kutta scheme. Such detection may include different properties, derived from physics, such as positivity, from numerics, such as a non-oscillatory behavior, or from computer requirements such as the absence of NaN's. Troubled cell values are discarded and re-computed starting again from the previous time-step using a more dissipative scheme but only locally, close to these cells. By locally decrementing the degree of the polynomial reconstructions from 2 to 0 we switch from a third-order to a first-order accurate but more stable scheme. The entropy indicator sensor is used to refine/coarsen the mesh. This sensor is also employed in an a posteriori manner because if some refinement is needed at the end of a time step, then the current time-step is recomputed with the refined mesh, but only locally, close to the new cells. We show on a large set of numerical tests that this a posteriori limiting procedure coupled with the entropy-based AMR technology can maintain not only optimal accuracy on smooth flows but also stability on discontinuous profiles such as shock waves, contacts, interfaces, etc. Moreover numerical evidences show that this approach is at least comparable in terms of accuracy and cost to a more classical CWENO approach within the same AMR context. [ABSTRACT FROM AUTHOR]

In this paper we deal with the extension of the Fast Kinetic Scheme (FKS) (Dimarco and Loubère, 2013 [26] ) originally constructed for solving the BGK equation, to the more challenging case of the Boltzmann equation. The scheme combines a robust and fast method for treating the transport part based on an innovative Lagrangian technique supplemented with conservative fast spectral schemes to treat the collisional operator by means of an operator splitting approach. This approach along with several implementation features related to the parallelization of the algorithm permits to construct an efficient simulation tool which is numerically tested against exact and reference solutions on classical problems arising in rarefied gas dynamic. We present results up to the 3 D × 3 D case for unsteady flows for the Variable Hard Sphere model which may serve as benchmark for future comparisons between different numerical methods for solving the multidimensional Boltzmann equation. For this reason, we also provide for each problem studied details on the computational cost and memory consumption as well as comparisons with the BGK model or the limit model of compressible Euler equations. [ABSTRACT FROM AUTHOR]

In this paper we propose a simple, robust and accurate nonlinear a posteriori stabilization of the Discontinuous Galerkin (DG) finite element method for the solution of nonlinear hyperbolic PDE systems on unstructured triangular and tetrahedral meshes in two and three space dimensions. This novel a posteriori limiter, which has been recently proposed for the simple Cartesian grid case in [62] , is able to resolve discontinuities at a sub-grid scale and is substantially extended here to general unstructured simplex meshes in 2D and 3D. It can be summarized as follows: At the beginning of each time step, an approximation of the local minimum and maximum of the discrete solution is computed for each cell, taking into account also the vertex neighbors of an element. Then, an unlimited discontinuous Galerkin scheme of approximation degree N is run for one time step to produce a so-called candidate solution . Subsequently, an a posteriori detection step checks the unlimited candidate solution at time t n + 1 for positivity, absence of floating point errors and whether the discrete solution has remained within or at least very close to the bounds given by the local minimum and maximum computed in the first step. Elements that do not satisfy all the previously mentioned detection criteria are flagged as troubled cells. For these troubled cells, the candidate solution is discarded as inappropriate and consequently needs to be recomputed . Within these troubled cells the old discrete solution at the previous time t n is scattered onto small sub-cells ( N s = 2 N + 1 sub-cells per element edge), in order to obtain a set of sub-cell averages at time t n . Then, a more robust second order TVD finite volume scheme is applied to update the sub-cell averages within the troubled DG cells from time t n to time t n + 1 . The new sub-grid data at time t n + 1 are finally gathered back into a valid cell-centered DG polynomial of degree N by using a classical conservative and higher order accurate finite volume reconstruction technique. Consequently, if the number N s is sufficiently large ( N s ≥ N + 1 ), the subscale resolution capability of the DG scheme is fully maintained, while preserving at the same time an essentially non-oscillatory behavior of the solution at discontinuities. Many standard DG limiters only adjust the discrete solution in troubled cells, based on the limiting of higher order moments or by applying a nonlinear WENO/HWENO reconstruction on the data at the new time t n + 1 . Instead, our new DG limiter entirely recomputes the troubled cells by solving the governing PDE system again starting from valid data at the old time level t n , but using this time a more robust scheme on the sub-grid level. In other words, the piecewise polynomials produced by the new limiter are the result of a more robust solution of the PDE system itself, while most standard DG limiters are simply based on a mere nonlinear data post-processing of the discrete solution. Technically speaking, the new method corresponds to an element-wise checkpointing and restarting of the solver, using a lower order scheme on the sub-grid. As a result, the present DG limiter is even able to cure floating point errors like NaN values that have occurred after divisions by zero or after the computation of roots from negative numbers. This is a unique feature of our new algorithm among existing DG limiters. The new a posteriori sub-cell stabilization approach is developed within a high order accurate one-step ADER-DG framework on multidimensional unstructured meshes for hyperbolic systems of conservation laws as well as for hyperbolic PDE with non-conservative products. The method is applied to the Euler equations of compressible gas dynamics, to the ideal magneto-hydrodynamics equations (MHD) as well as to the seven-equation Baer–Nunziato model of compressible multi-phase flows. A large set of standard test problems is solved in order to assess the accuracy and robustness of the new limiter. [ABSTRACT FROM AUTHOR]

In this paper we consider the extension of the method developed in Dimarco and Loubère (2013) [22,23] with the aim of facing the numerical resolution of multi-scale problems arising in rarefied gas dynamics. The scope of this work is to consider situations in which the whole domain does not demand the use of a kinetic model everywhere. This is the case of many realistic applications: some regions of the computational domain require a microscopic description, given by a kinetic model while the rest of the domain can be described by a coarser model of fluid type. Our aim is to show how the kinetic scheme developed in the pre-cited articles is perfectly suited for building domain decomposition strategies which make the method more attractive with respect to classical numerical techniques for kinetic equations and multi-scale realistic problems. Several numerical evidences are provided in this work in the two dimensional and three dimensional settings to assess the efficiency of the domain decomposition scheme. [ABSTRACT FROM AUTHOR]

In this paper we demonstrate the capability of the fast semi-Lagrangian scheme developed in [20] and [21] to deal with parallel architectures. First, we will present the behaviors of such scheme on a classical architecture using OpenMP and then on GPU (Graphics Processing Unit) architecture using CUDA. The goal is to prove that this new scheme is well adapted to these types of parallelizations, and, moreover that the gain in CPU time is substantial on nowadays affordable computers. We first present the sequential version of our high-order kinetic scheme and focus on important details for an effective parallel implementation. Then, we introduce the specific treatments and algorithms which have been developed for an OpenMP and CUDA parallelizations. Numerical tests are shown for the full 3D/3D simulations. These assess the important speed-up factor of the method gained between the sequential code and the parallel versions and its very good scalability which makes this approach a real competitor with respect to existing schemes for the solution of multidimensional kinetic models. [ABSTRACT FROM AUTHOR]

Abstract: We present a new reconnection-based arbitrary-Lagrangian–Eulerian (ALE) method. The main elements in a standard ALE simulation are an explicit Lagrangian phase in which the solution and grid are updated, a rezoning phase in which a new grid is defined, and a remapping phase in which the Lagrangian solution is transferred (conservatively interpolated) onto the new grid. In standard ALE methods the new mesh from the rezone phase is obtained by moving grid nodes without changing connectivity of the mesh. Such rezone strategy has its limitation due to the fixed topology of the mesh. In our new method we allow connectivity of the mesh to change in rezone phase, which leads to general polygonal mesh and allows to follow Lagrangian features of the mesh much better than for standard ALE methods. Rezone strategy with reconnection is based on using Voronoi tessellation. We demonstrate performance of our new method on series of numerical examples and show it superiority in comparison with standard ALE methods without reconnection. [Copyright &y& Elsevier]

Abstract: The repair paradigm leads to several algorithms for redistributing mass, momentum and energy, while adhering to local maximum principles, as an adjunct to the remapping step in certain compressible flow codes that use remapping, such as Arbitrary–Lagrangian–Eulerian codes, or for just redistributing mass in advection codes. In the case of advection of a concentration, repair keeps the newly computed concentration in a cell between the maximum and minimum concentrations in neighboring old cells, thus guaranteeing at least that the new concentration is between zero and one. For compressible flow, density, velocity and internal energy are similarly constrained while maintaining conservation of mass, momentum and total energy. In this way, positive density and internal energy are achieved as a side effect. We propose a new algorithm, combining both local and global repair, that maintains causality and is efficient in a parallel computational setting. The local/global algorithm is independent of the order in which the distribution is performed, and it maintains 1D symmetry. This is applied to advection in two dimensions, and to, among others, the LeBlanc problem, the Sedov problem, and an interacting 2D blast wave problem. The latter is done with a Lagrangian code for which rezoning, remapping and repair are essential. [Copyright &y& Elsevier]

Abstract: We describe a new remapping algorithm for use in arbitrary Lagrangian–Eulerian (ALE) simulations. The new features of this remapper are designed to complement a staggered-mesh Lagrangian phase in which the cells may be general polygons (in two dimensions), and which uses subcell discretizations to control unphysical mesh distortion and hourglassing. Our new remapping algorithm consists of three stages. A gathering stage, in which we interpolate momentum, internal energy, and kinetic energy to the subcells in a conservative way. A subcell remapping stage, in which we conservatively remap mass, momentum, internal, and kinetic energy from the subcells of the Lagrangian mesh to the subcells of the new rezoned mesh. A scattering stage, in which we conservatively recover the primary variables: subcell density, nodal velocity, and cell-centered specific internal energy on the new rezoned mesh. We prove that our new remapping algorithm is conservative, reversible, and satisfies the DeBar consistency condition. We also demonstrate computationally that our new remapping method is robust and accurate for a series of test problems in one and two dimensions. [Copyright &y& Elsevier]

• Derivation of 3D hyperelasticity model and governing equations. • New updated Lagrangian finite volume scheme for hyperelasticity. • Space-time second order accurate scheme with a posteriori MOOD limiting. • Unstructured 3D moving meshes. • Geometric Conservation Law compliant discretization. In this paper, we present a conservative cell-centered Lagrangian Finite Volume scheme for solving the hyperelasticity equations on unstructured multidimensional grids. The starting point of the present approach is the cell-centered FV discretization named EUCCLHYD and introduced in the context of Lagrangian hydrodynamics. Here, it is combined with the a posteriori Multidimensional Optimal Order Detection (MOOD) limiting strategy to ensure robustness and stability at shock waves with piecewise linear spatial reconstruction. The ADER (Arbitrary high order schemes using DERivatives) approach is adopted to obtain second-order of accuracy in time. This strategy has been successfully tested in a hydrodynamics context and the present work aims at extending it to the case of hyperelasticity. Here, the hyperelasticity equations are written in the updated Lagrangian framework and the dedicated Lagrangian numerical scheme is derived in terms of nodal solver, Geometrical Conservation Law (GCL) compliance, subcell forces and compatible discretization. The Lagrangian numerical method is implemented in 3D under MPI parallelization framework allowing to handle genuinely large meshes. A relatively large set of numerical test cases is presented to assess the ability of the method to achieve effective second order of accuracy on smooth flows, maintaining an essentially non-oscillatory behavior and general robustness across discontinuities and ensuring at least physical admissibility of the solution where appropriate. Pure elastic neo-Hookean and non-linear materials are considered for our benchmark test problems in 2D and 3D. These test cases feature material bending, impact, compression, non-linear deformation and further bouncing/detaching motions. [ABSTRACT FROM AUTHOR]

Implicit time discretization in computational fluid dynamics dedicated to compute steady state solution of hypersonic flows was an intense field of research in the 70's-80's. It is suitable for computational efficiency to use implicit schemes that do not suffer from time step restriction to guarantee stability, unlike explicit ones. Unfortunately time step restriction is still required in practice, especially for stiff numerical test cases such as high Mach number flows around objects. A method introduced by Yee et al. (1985) [34] is commonly used to simulate computational fluid dynamics problems in an implicit fashion. However this method has no formal theoretical basis for systems of conservation laws. Consequently the practical time step is driven by a ad-hoc user-given profile. The purpose of this work is first to study the mathematical properties of such linearized implicit finite volume schemes to enlighten their weaknesses and exhibit more adequate linearization processes. We rely on the hyperbolicity of the Euler equations to establish a general framework to design implicit schemes. Secondly, we propose a correction of the system matrix to adapt to any given finite volume scheme. The obtained linearized implicit finite volume methods are more robust and less constrained with regards to ad-hoc user-given profile. Numerical results in 1D and 2D will provide evidences to confirm the analysis on relevant and challenging hypersonic test cases. • Implicit schemes are suitable to compute steady state solution of hypersonic flows. • Theoretically, no time step restriction needed to ensure implicit schemes stability. • Study of the mathematical properties of linearized implicit finite volume schemes. • Novel mathematical framework to design more robust linearized implicit schemes. • Correction of the matrix to adapt implicit scheme to any given finite volume scheme. [ABSTRACT FROM AUTHOR]

In this paper we blend two lattice-Boltzmann (LB) numerical schemes with an a posteriori Multi-dimensional Optimal Order Detection (MOOD) paradigm to solve hyperbolic systems of conservation laws in 1D and 2D. The first LB scheme is robust to the presence of shock waves but lacks accuracy on smooth flows. The second one has a second-order of accuracy but develops non-physical oscillations when solving steep gradients. The MOOD paradigm produces a hybrid LB scheme via smooth and positivity detectors allowing to gather the best properties of the two LB methods within one scheme. Indeed, the resulting scheme presents second order of accuracy on smooth solutions, essentially non-oscillatory behaviour on irregular ones, and, an 'almost fail-safe' property concerning positivity issues. The numerical results on a set of sanity test cases and demanding ones are presented assessing the appropriate behaviour of the hybrid LBMOOD scheme in 1D and 2D. [ABSTRACT FROM AUTHOR]

In this paper we blend two lattice-Boltzmann (LB) numerical schemes with an a posteriori Multi-dimensional Optimal Order Detection (MOOD) paradigm to solve hyperbolic systems of conservation laws in 1D and 2D. The first LB scheme is robust to the presence of shock waves but lacks accuracy on smooth flows. The second one has a second-order of accuracy but develops non-physical oscillations when solving steep gradients. The MOOD paradigm produces a hybrid LB scheme via smooth and positivity detectors allowing to gather the best properties of the two LB methods within one scheme. Indeed, the resulting scheme presents second order of accuracy on smooth solutions, essentially non-oscillatory behaviour on irregular ones, and, an 'almost fail-safe' property concerning positivity issues. The numerical results on a set of sanity test cases and demanding ones are presented assessing the appropriate behaviour of the hybrid LBMOOD scheme in 1D and 2D. [ABSTRACT FROM AUTHOR]

• Theoretical comparison between hypoelastic and hyperelastic description of the dynamics of solids. • Wilkins model for hypoelasticity and Godunov-Peshkov-Romenski model for hyperelasticity. • Numerical comparison of both model using FV and DG schemes on structured and unstructured meshes. The aim of this paper is to compare a hyperelastic with a hypoelastic model describing the Eulerian dynamics of solids in the context of non-linear elastoplastic deformations. Specifically, we consider the well-known hypoelastic Wilkins model, which is compared against a hyperelastic model based on the work of Godunov and Romenski. First, we discuss some general conceptual differences of the two approaches. Second, a detailed study of both models is proposed, where differences are made evident at the aid of deriving a hypoelastic-type model corresponding to the hyperelastic model and a particular equation of state used in this paper. Third, using the same high order ADER Finite Volume and Discontinuous Galerkin methods on fixed and moving unstructured meshes for both models, a wide range of numerical benchmark test problems has been solved. The numerical solutions obtained for the two different models are directly compared with each other. For small elastic deformations, the two models produce very similar solutions that are close to each other. However, if large elastic or elastoplastic deformations occur, the solutions present larger differences. [ABSTRACT FROM AUTHOR]

In this paper we develop a conservative cell-centered Lagrangian finite volume scheme for the solution of the hydrodynamics equations on unstructured multidimensional grids. The method is derived from the Eucclhyd scheme discussed in [47,43,45] . It is second-order accurate in space and is combined with the a posteriori Multidimensional Optimal Order Detection (MOOD) limiting strategy to ensure robustness and stability at shock waves. Second-order of accuracy in time is achieved via the ADER (Arbitrary high order schemes using DERivatives) approach. A large set of numerical test cases is proposed to assess the ability of the method to achieve effective second order of accuracy on smooth flows, maintaining an essentially non-oscillatory behavior on discontinuous profiles, general robustness ensuring physical admissibility of the numerical solution, and precision where appropriate. [ABSTRACT FROM AUTHOR]

This paper describes a novel subface flux-based Finite Volume (FV) method for discretizing multi-dimensional hyperbolic systems of conservation laws of general unstructured grids. The subface flux numerical approximation relies on the notion of simple Eulerian Riemann solver introduced in the seminal work [Gallice (2003) [18] ]. The Eulerian Riemann solver is constructed from its Lagrangian counterpart by means of the Lagrange-to-Euler mapping. This systematic procedure ensures the transfer of good properties such as positivity preservation and entropy stability. In this framework, the conservativity and the entropy stability are no more locally face-based but result respectively from a node-based vectorial equation and a scalar inequation. The corresponding multi-dimensional FV scheme is characterized by an explicit time step condition ensuring positivity preservation and entropy stability. The application to gas dynamics provides an original multi-dimensional conservative and entropy-stable FV scheme wherein the numerical fluxes are computed through a nodal solver which is similar to the one designed for Lagrangian hydrodynamics. The robustness and the accuracy of this novel FV scheme are assessed through various numerical tests. We observe its insensitivity to the numerical pathologies that plague classical face-based contact discontinuity preserving FV formulations. • A novel subface flux-based Finite Volume (FV) discretization for hyperbolic systems. • Subface flux approximation relies on Eulerian simple Riemann solvers. • Eulerian solvers deduced systematically from their Lagrangian counterparts. • Conservation and entropy stability ensured through node-based conditions. • A FV method insensitive to the classical numerical pathologies such as carbuncle. [ABSTRACT FROM AUTHOR]

The purpose of this work is to propose a novel a posteriori finite volume subcell limiter technique for the Discontinuous Galerkin finite element method for nonlinear systems of hyperbolic conservation laws in multiple space dimensions that works well for arbitrary high order of accuracy in space and time and that does not destroy the natural subcell resolution properties of the DG method. High order time discretization is achieved via a one-step ADER approach that uses a local space–time discontinuous Galerkin predictor method to evolve the data locally in time within each cell. Our new limiting strategy is based on the so-called MOOD paradigm, which a posteriori verifies the validity of a discrete candidate solution against physical and numerical detection criteria after each time step. Here, we employ a relaxed discrete maximum principle in the sense of piecewise polynomials and the positivity of the numerical solution as detection criteria. Within the DG scheme on the main grid, the discrete solution is represented by piecewise polynomials of degree N . For those troubled cells that need limiting, our new limiter approach recomputes the discrete solution by scattering the DG polynomials at the previous time step onto a set of N s = 2 N + 1 finite volume subcells per space dimension. A robust but accurate ADER-WENO finite volume scheme then updates the subcell averages of the conservative variables within the detected troubled cells. The recomputed subcell averages are subsequently gathered back into high order cell-centered DG polynomials on the main grid via a subgrid reconstruction operator. The choice of N s = 2 N + 1 subcells is optimal since it allows to match the maximum admissible time step of the finite volume scheme on the subgrid with the maximum admissible time step of the DG scheme on the main grid, minimizing at the same time also the local truncation error of the subcell finite volume scheme. It furthermore provides an excellent subcell resolution of discontinuities. Our new approach is therefore radically different from classical DG limiters, where the limiter is using TVB or (H)WENO reconstruction based on the discrete solution of the DG scheme on the main grid at the new time level. In our case, the discrete solution is recomputed within the troubled cells from the old time level using a different and more robust numerical scheme on a subgrid level . We illustrate the performance of the new a posteriori subcell ADER-WENO finite volume limiter approach for very high order DG methods via the simulation of numerous test cases run on Cartesian grids in two and three space dimensions, using DG schemes of up to tenth order of accuracy in space and time ( N = 9 ). The method is also able to run on massively parallel large scale supercomputing infrastructure, which is shown via one 3D test problem that uses 10 billion space–time degrees of freedom per time step. [ABSTRACT FROM AUTHOR]

• All Mach solver for the Euler equations. • Asymptotic Preserving discretization. • Fully second order space-time method. • Collocated numerical scheme on a 3D Cartesian grid. This article deals with the development of a numerical method for the compressible Euler system valid for all Mach numbers: from extremely low to high regimes. In classical fluid dynamic problems, one faces both situations in which the flow is subsonic, and consequently acoustic waves are very fast compared to the velocity of the fluid, and situations in which the fluid moves at high speed and compressibility may generate shock waves. Standard explicit fluid solvers such as Godunov method fail in the description of both flows due to time step restrictions caused by the stiffness of the equations which leads to prohibitive computational costs. In this work, we develop a new method for the full Euler system of gas dynamics based on partitioning the equations into a fast and a low scale. Such a method employs a time step which is independent of the speed of the pressure waves and works uniformly for all Mach numbers. Cell centered discretization on Cartesian meshes is proposed. Numerical results up to the three dimensional case show the accuracy, the robustness and the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

In this article, we present a multi-dimensional-aware Eulerian Riemann Solver (RS) and its associated Finite Volume (FV) scheme for the 2D Shallow-Water (SW) equations. This RS, appropriately derived from its associated Lagrangian version, presents the specific feature of coupling all cells in the vicinity of the current one. Consequently, this solver is no longer a 1D RS across one edge. Contrarily, it encounters for genuine multidimensional effects and for the presence of the source term of the SW equations. The associated first order FV numerical scheme ensures well-balancing for lake at rest steady states, positivity preservation and entropy stability properties. Moreover, a second-order accurate extension is proposed based on Runge-Kutta time discretization and piecewise linear limited reconstructions, that preserve the well-balanced character of the first order scheme. We present several 2D tests assessing the good behaviors of the obtained numerical scheme on unstructured mesh. The numerical scheme seems insensitive to spurious numerical instabilities such as the carbuncle effect. • Subface-based multi-dimensional-aware finite volume scheme on unstructured grids. • Scheme for conservation laws extended to balance laws: shallow water as a prototype. • Positivity- and entropy-stability properties generalize to the case of balance laws. • Well-balanced Eulerian Riemann solver derived from its Lagrangian counterpart. • A scheme insensible to instabilities such as Carbuncle and odd-even decoupling. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we describe a cell-centered Lagrangian scheme devoted to the numerical 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 deviatoric 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 Falcovitz [M. Ben-Artzi, J. Falcovitz, Generalized Riemann Problems in Computational Fluid Dynamics, Cambridge Monogr. Appl. Comput. Math. (2003)]. Finally, the robustness and the accuracy of the numerical scheme are assessed through the computation of several test cases. [Copyright &y& Elsevier]

Abstract: A new, second-order accurate, volume conservative, material-order-independent interface reconstruction method for multi-material flow simulations is presented. First, materials are located in multi-material computational cells using a piecewise linear reconstruction of the volume fraction function. These material locator points are then used as generators to reconstruct the interface with a weighted Voronoi diagram that matches the volume fractions. The interfaces are then improved by minimizing an objective function that smoothes interface normals while enforcing convexity and volume constraints for the pure material subcells. Convergence tests are shown demonstrating second-order accuracy. Static and dynamic examples are shown illustrating the superior performance of the method over existing material-order-dependent methods. [Copyright &y& Elsevier]