Raphaël Loubère

• PhD Applied Mathematics -Habilitation Applied Mathematics
• Research Director at French National Centre for Scientific Research

Instead of ensuring that fluxes across edges add up to zero, we split the edge in two halves and also associate different fluxes to each of its sides. This is possible due to non-standard Riemann solvers with free parameters. We then enforce conservation by making sure that the fluxes around a node sum up to zero, which fixes the value of the free...

Instead of ensuring that fluxes across edges add up to zero, we split the edge in two halves and also associate different fluxes to each of its sides. This is possible due to non-standard Riemann solvers with free parameters. We then enforce conservation by making sure that the fluxes around a node sum up to zero, which fixes the value of the free...

We construct an unconventional divergence preserving discretization of updated Lagrangian ideal magnetohydrodynamics (MHD) over simplicial grids. The cell-centered finite-volume (FV) method employed to discretize the conservation laws of volume, momentum, and total energy is rigorously the same as the one developed to simulate hyperelasticity equat...

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,...

In this paper we blend high-order Compact Approximate Taylor (CAT) numerical methods with the a posteriori Multi-dimensional Optimal Order Detection (MOOD) paradigm to solve hyperbolic systems of conservation laws. The resulting methods are highly accurate for smooth solutions, essentially non-oscillatory for discontinuous ones, and almost fail-saf...

In this paper we blend high-order Compact Approximate Taylor (CAT) numerical methods with the a posteriori Multi-dimensional Optimal Order Detection (MOOD) paradigm to solve hyperbolic systems of conservation laws. The resulting methods are highly accurate for smooth solutions, essentially non-oscillatory for discontinuous ones, and almost fail-saf...

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 [G. Gallice; Positive and entropy stable Go...

Considering transient processes where liquid/solid phase change occurs, this paper focuses on the associated modeling and numerical treatment in the frame of “Computational Fluid Dynamics” simulations. While being of importance in many industrial applications involving solidification and melting of mixed materials, including power and manufacturing...

Considering phase changes associated with a high-temperature molten material cooled down from the outside, this work presents an improvement of the modelling and the numerical simulation of such processes for an application pertaining to the safety of light water nuclear reactors. Postulating a core meltdown accident, the behaviour of the core melt...

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...

We propose an adaptive stencil construction for high-order accurate finite volume schemes a posteriori stabilized devoted to solve one-dimensional steady-state hyperbolic equations. High accuracy (up to the sixth-order presently) is achieved, thanks to polynomial reconstructions while stability is provided with an a posteriori MOOD method which con...

This talk presents the development of FV methods to solve the non-viscous and non-conductive part of the N-S equations, i.e. the two-dimensional Euler equations. It focuses on unstructured meshes and use the Lagrangian framework as a stepping stone to derive an approximate RS in the Eulerian framework. On the ground of this, direct estimations of o...

In this paper we propose to revisit the notion of simple Riemann solver in Lagrangian coordinates follow- ing Gallice ”Positive and entropy stable godunov-type schemes for gas dynamics and MHD equations in Lagrangian or Eulerian coordinates” in Numer. Math., 94, 2003. We interpret and supplement this work with an advanced study on the notions of wa...

In this paper we present a conservative cell-centered Lagrangian finite volume scheme for the solution of the hyper-elasticity equations on unstructured multidimensional grids. The starting point of the new method is the Eucclhyd scheme, which is here combined with the a posteriori Multidimensional Optimal Order Detection (MOOD) limiting strategy t...

We propose an adaptive stencil construction for high order accurate finite volume schemes aposteriori stabilized devoted to solve one-dimensional steady-state hyperbolic equations. High-accuracy (up to the sixth-order presently) is achieved thanks to polynomial reconstructions while stability is provided with an aposteriori MOOD method which contro...

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 [G. Gallice; Positive and entropy stable Go...

In this work we present an attempt to replace an a posteriori MOOD loop used in a high accurate Finite Volume (FV) scheme by a trained artificial Neural Network (NN). The MOOD loop, by decrementing the reconstruction polynomial degrees, ensures accuracy, essentially non-oscillatory, robustness properties and preserves physical features. Indeed it r...

In this article we present a 1D single-material conservative remapping method that relies on high accurate reconstructions: polynomial (P4, P1 with slope limiter) and non-linear hyperbolic tangent (THINC) representations. Such remapping procedure is intended to be used pairwise with a cell-centered Lagrangian scheme along with a rezone strategy to...

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 discon...

In this paper we propose a novel semi-implicit Discontinuous Galerkin (DG) finite element scheme on staggered meshes with a posteriori subcell finite volume limiting for the one and two dimensional Euler equations of compressible gasdynamics. We therefore extend the strategy adopted by Dumbser and Casulli (Appl Math Comput 272:479–497, 2016), where...

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 situati...

The purpose of this work is to build a general framework to reconstruct the underlying fields within a Finite Volume (FV) scheme solving a hyperbolic system of PDEs (Partial Differential Equations). In an FV context, the data are piece-wise constants per computational cell and the physical fields are reconstructed taking into account neighbor cell...

Develop a genuinely non-linear BVD FV scheme with a posteriori MOOD loop in 1D and 2D.

Presentation given in september 2019 at Trento Multimat conférence

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 discus...

We introduce an extension of the fast semi-Lagrangian scheme developed in J Comput Phys 255:680–698 (2013) in an effort to increase the spatial accuracy of the method. The basic idea of this extension is to modify the first-order accurate transport step of the original semi-Lagrangian scheme to allow for a general piecewise polynomial reconstructio...

Obtaining very high‐order accurate solutions in curved domains is a challenging task as the accuracy of discretization methods may dramatically reduce without an appropriate treatment of boundary conditions. The classical techniques to preserve the nominal convergence order of accuracy, proposed in the context of finite element and finite volume me...

We consider the well-balanced numerical scheme for the shallow water equations with topography introduced in Michel-Dansac et al. (Comput Math Appl 72(3):568–593, 2016, [8]) and its second-order well-balanced extension, which requires two heuristic parameters. The goal of the present contribution is to derive a parameter-free second-order well-bala...

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 discus...

This work proposes a new spatial reconstruction scheme in finite volume frameworks. Different from long-lasting reconstruction processes which employ high order polynomials enforced with some carefully designed limiting projections to seek stable solutions around discontinuities, the current discretized scheme employs THINC (Tangent of Hyperbola fo...

We propose a new family of high order accurate finite volume schemes devoted to solve one-dimensional steady-state hyperbolic systems. High-accuracy (up to the sixth-order presently) is achieved thanks to polynomial reconstructions while stability is provided with an a posteriori MOOD method which controls the cell polynomial degree for eliminating...

During typesetting, Figs. 8 and 21 got corrupted and the images shown in the online published version are not correct. The original publication was updated.

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...

Accuracy may be dramatically reduced when the boundary domain is curved and numerical schemes require a specific treatment of the boundary condition to preserve the optimal order. In the finite volume context, Ollivier-Gooch and Van Altena (2002) has proposed a technique to overcome such limitation and restore the very high-order accuracy which con...

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 stag...

In this work, we consider the development of implicit explicit total variation diminishing (TVD) methods (also termed SSP: strong stability preserving) for the compressible isentropic Euler system in the low Mach number regime. The scheme proposed is asymptotically stable with a CFL condition independent from the Mach number and it degenerates in t...

In this work, we consider the development of implicit explicit total variation diminishing (TVD) methods (also termed SSP: strong stability preserving) for the compressible isentropic Euler system in the low Mach number regime. The scheme proposed is asymptotically stable with a CFL condition independent from the Mach number and it degenerates in t...

Hourglassing is a well-known pathological numerical artifact affecting the robustness and accuracy of Lagrangian methods. There exist a large number of hourglass control/suppression strategies. In the community of the staggered compatible Lagrangian methods, the approach of sub-zonal pressure forces is among the most widely used. However, this appr...

In this article we show the gain in accuracy and robustness brought by the use of a a posteriori MOOD limiting in replacement of the classical slope limiter employed in the remap phase of a legacy second-order Lagrange+Remap scheme solving the Euler system of equations. This simple substitution ensures extended robustness property, better accuracy...

We propose a sixth-order staggered finite volume scheme based on polynomial reconstructions to achieve high accurate numerical solutions for the incompressible Navier–Stokes and Euler equations. The scheme is equipped with a fixed-point algorithm with solution relaxation to speed-up the convergence and reduce the computation time. Numerical tests a...

Hourglassing is a well-known pathological numerical artifact affecting the ro-bustness and accuracy of Lagrangian methods. There exist a large number of hourglass control/suppression strategies. In the community of the staggered compatible Lagrangian methods, the approach of sub-zonal pressure forces is among the most widely used. However, this app...

In this paper we present a 2D/3D high order accurate finite volume scheme in the context of direct Arbitrary-Lagrangian-Eulerian algorithms for general hyperbolic systems of partial differential equations with non-conservative products and stiff source terms. This scheme is constructed with a single stencil polynomial reconstruction operator, a one...

In this paper we deal with the extension of the Fast Kinetic Scheme (FKS) [J. Comput. Phys., Vol. 255, 2013, pp 680-698] 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...

In this paper we deal with the extension of the Fast Kinetic Scheme (FKS) [J. Comput. Phys., Vol. 255, 2013, pp 680-698] 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...

We present a new high-accurate, stable and low-dissipative Smooth Particle Hydrodynamics (SPH) method based on Riemann solvers. The method derives from the SPH-ALE formulation first proposed by Vila and Ben Moussa. Moving Least Squares approximations are used for the reconstruction of the variables and the computation of Taylor expansions. The stab...

In this article we present a high order accurate 2D conservative remapping method for a general polygonal mesh. This method conservatively projects piecewise constant data from an old mesh onto a possibly uncorrelated new one. First an arbitrary (high) accuracy polynomial reconstruction operator is built. Then, the exact intersection between the ol...

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. Subseque...

This article deals with the discretization of the compressible Euler system for all Mach numbers regimes. For highly subsonic flows, since acoustic waves are very fast compared to the velocity of the fluid, the gas can be considered as incompressible. From the numerical point of view, when the Mach number tends to zero, the classical Godunov type s...

In this paper we genealize the fast semi-Lagrangian scheme developed in [J. Comput. Phys., Vol. 255, 2013, pp 680-698] to the case of high order reconstructions of the distribution function. The original first order accurate semi-Lagrangian scheme is supplemented with polynomial reconstructions of the distribution function and of the collisional op...

This paper is concerned with the numerical solution of the unified first order hyperbolic formulation of continuum mechanics proposed by Peshkov & Romenski (HPR model), which is based on the theory of nonlinear hyperelasticity of Godunov & Romenski . Notably, the governing PDE system is symmetric hyperbolic and fully consistent with the first and t...

This paper is concerned with the numerical solution of the unified first order hyperbolic formulation of continuum mechanics proposed by Peshkov & Romenski (HPR model), which is based on the theory of nonlinear hyperelasticity of Godunov & Romenski . Notably, the governing PDE system is symmetric hyperbolic and fully consistent with the first and t...

We present the two main types of Finite Volume Lagrangian schemes named: staggered-grid hydrodynamics (SGH) and colocated Lagrangian hydrodynamics (CLH). Both are devoted to solve the hydrodynamic conservation laws and extended system in multidimension on general grid. They are funded on common paradigms, such as the need to solve the conservation...

In this article we present a 2D conservative remapping method which relies on exact polygonal mesh in-tersection, high accurate polynomial reconstruction (up to degree 5) and a posteriori stabilization based onMOOD paradigm [21, 30, 31, 80]. This paradigm does not compute any sort of a priori limiter for the poly-nomial reconstructions. Instead it...

In this paper we present a new family of efficient high order accurate direct Arbitrary-Lagrangian–Eulerian (ALE) one-step ADER-MOOD finite volume schemes for the solution of nonlinear hyperbolic systems of conservation laws for moving unstructured triangular and tetrahedral meshes. This family is the next generation of the ALE ADER-WENO schemes pr...

In this paper we consider the extension of the method developed in Dimarco and Loubère (2013) [22] and [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 i...

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 w...

In this work, we propose an adaptive subdivision piecewise linear interface calculation (PLIC) for 2D multimaterial hydrodynamic simulation codes. Classical volume-of-fluid PLIC technique uses one line segment and one given normal to separate two materials. Unfortunately, these paradigms are not sufficient when filaments occur, leading to the creat...

In this paper, we investigate the coupling of the Multi-dimensional Optimal Order Detection (MOOD) method and the Arbitrary high order DERivatives (ADER) approach in order to design a new high order accurate, robust and computationally efficient Finite Volume (FV) scheme dedicated to solve nonlinear systems of hyperbolic conservation laws on unstru...

The purpose of this work is to propose a novel a posteriori finite volume subcell limitation 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, preserving the natural subcell resoluti...

This work concerns the simulation of compressible multimaterial fluid flows and follows up the method Finite Volume with Characteristics Flux for two materials described in paper [R. Loubère, J.P. Braeunig, J.-M Ghidaglia, A totally Eulerian finite volume solver for multimaterial fluid flows: enhanced Natural Interface Positioning (ENIP), Eur. J. o...

The Multidimensional Optimal Order Detection (MOOD) method for two-dimensional geometries has been introduced by the authors in two recent papers. We present here the extension to 3D mixed meshes composed of tetrahedra, hexahedra, pyramids, and prisms. In addition, we simplify the u2 detection process previously developed and show on a relevant set...

In this paper, we investigate the coupling of the Multi-dimensional Optimal Order De- tection (MOOD) method and the Arbitrary high order DERivatives (ADER) approach in order to design a new high order accurate, robust and computationally efficient Finite Volume (FV) scheme dedicated to solve nonlinear systems of hyperbolic conservation laws on unst...

This thesis presents our work related to (i) Lagrangian schemes and (ii) Arbitrary- Lagrangian-Eulerian numerical methods (ALE). Both types of methods have in commun to solve the multidimension compressible Euler equations on a moving grid. The grid moves with either the fluid velocity (Lagrangian) or an arbitrary velocity (ALE). More specifically...

In this paper, we investigate an original way to deal with the problems generated by the limitation process of high-order finite volume methods based on polynomial reconstructions. Multi-dimensional Optimal Order Detection (MOOD) breaks away from classical limitations employed in high-order methods. The proposed method consists of detecting problem...

SUMMARY The aim of the present work is the 3D extension of a general formalism to derive a staggered discretization for Lagrangian hydrodynamics on unstructured grids. The classical compatible discretization is used; namely, momentum equation is discretized using the fundamental concept of subcell forces. Specific internal energy equation is obtain...

In a recent paper we presented a new ultra efficient numerical method for solving kinetic equations of the Boltzmann type (G. Dimarco, R. Loubere, Towards an ultra efficient kinetic scheme. Part I: basics on the 689 BGK equation, J. Comp. Phys., (2013), http://dx.doi.org/10.1016/j.jcp.2012.10.058). The key idea, on which the method relies, is to so...

A finite volume cell-centered Lagrangian scheme for solving large deformation problems is constructed based on the hypo-elastic model. Since solid materials can sustain significant shear deformation, evolution equations for stress and strain fields are solved in addition to mass, momentum and energy conservation laws. In order to evolve the momentu...

The Multi-dimensional Optimal Order Detection (MOOD) method for two-dimensional geometries has been introduced in "A high-order finite volume method for hyperbolic systems: Multi-dimensional Optimal Order Detection (MOOD)", J. Comput. Phys. 230 (2011), and enhanced in "Improved Detection Criteria for the Multi-dimensional Optimal Order Detection (M...

This paper extends the MOOD method proposed by the authors in [A high-order finite volume method for hyperbolic systems: Multi-Dimensional Optimal Order Detection (MOOD). J Comput Phys 2011;230:4028–50], along two complementary axes: extension to very high-order polynomial reconstruction on non-conformal unstructured meshes and new detection criter...

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. Co...

The Multi-dimensional Optimal Order Detection (MOOD) method has been designed by authors in [5] and extended in [7] to reach Very-High-Order of accuracy for systems of Conservation Laws in a Finite Volume (FV) framework on 2D unstructured meshes. In this paper we focus on the extension of this method to 3D unstructured meshes. We present preliminar...

In this paper we present a new ultra efficient numerical method for solving kinetic equations. In this preliminary work, we present the scheme in the case of the BGK relaxation operator. The scheme, being based on a splitting technique between transport and collision, can be easily extended to other collisional operators as the Boltzmann collision...

Many hydrodynamical problems involve shear flows along material interfaces. If the materials move along each other but are tied to a single Lagrangian computational mesh without any sliding treatment, severe mesh distortions appear which can eventually cause the failure of the simulation. This problem is usually treated by introducing the sliding l...

We propose a new finite volume method to provide very high-order accuracy for the convection diffusion problem. The main tool is a polynomial reconstruction based on the mean-value to provide the best order. We give simple numerical examples that illustrate the effectiveness of the method in attaining the expected order of convergence.

The aim of the present work is to develop a general formalism to derive staggered discretizations for Lagrangian hydrodynamics on two-dimensional unstructured grids. To this end,wemake use of the compatible discretization that has been initially introduced by E. J. Caramana et al., in J. Comput. Phys., 146 (1998). Namely, momentum equation is discr...

This paper deals with the extension to the cylindrical geometry of the recently introduced Reconnection algorithm for Arbitrary-Lagrangian–Eulerian (ReALE) framework. The main elements in standard ALE methods are an explicit Lagrangian phase, a rezoning phase, and a remapping phase. Usually the new mesh provided by the rezone phase is obtained by m...

The Multi-dimensional Optimal Order Detection (MOOD) method is an original Very High-Order Finite Volume (FV) method for conservation laws on unstructured meshes. The method is based on an a posteriori degree reduction of local polynomial reconstructions on cells where prescribed stabil-ity conditions are not fulfilled. Numerical experiments on adv...

In this paper, we investigate a new way to deal with the problems generated by the limitation process of very high-order finite volume methods based on polyno- mial reconstructions. Multi-dimensional Optimal Order Detection (MOOD) breaks away from classical limitations employed in MUSCL or ENO/WENO. Indeed, instead of classical limiting of polynomi...

This work concerns the simulation of compressible multi-material fluid flows and follows the method FVCF-NIP described in the former paper Braeunig et al (Eur. J. Mech. B/Fluids, 2009). This Cell-centered Finite Volume method is totally Eulerian since the mesh is not moving and a sharp interface, separating two materials, evolves through the grid....