An entropy preserving relaxation scheme for ten-moments equations with source terms

International audienceThe present paper concerns the derivation of finite volume methods to approximate weak solutions of Ten-Moments equations with source terms. These equations model compressible anisotropic flows. A relaxation-type scheme is proposed to approximate such flows. Both robustness and stability conditions of the suggested finite volume methods are established. To prove discrete entropy inequalities, we derive a new strategy based on local minimum entropy principle and never use some approximate PDE's auxiliary model as usually recommended. Moreover, numerical simulations in 1D and in 2D illustrate our approach

Derivation and numerical approximation of two-temperature Euler plasma model

International audienceThis paper gives a derivation of the two-temperature Euler plasma system from the two-fluid MHD model. The two-temperature Euler plasma system is proved to be an asymptotic regime for weakly magnetized plasma of the two-fluid MHD model. Our procedure is more general, enabling us to show that assumptions in previous derivations in literature are straightforward consequences of our work. We then propose a finite volume approximation to compute the solution of the two-temperature Euler plasma model in unstructured tessellations used to adequately mesh the toroidal geometry of the tokamak, where flows the plasma. Numerical tests illustrate our method

Finite Volume Approximation of MHD Equations with Euler Potential

International audienceThe possibility to produce energy by fusion reactions is being studied in experimental devices called tokamaks where charged particles are confined in a toroidal vacuum chamber thanks to a very large magnetic field. The ITER device currently build in Cadarache (France) will be the largest tokamak ever realized for these experiments. The large scale dynamics of charged particles in a tokamak as ITER can be described by the Magneto-HydroDynamics equations (MHD). This system of equations contains as an involution the divergence-free constraint of the magnetic field, div B = 0 that has to be maintained by the numerical approximation. The respect of the divergence-free constraint can be achieved in two different ways. The first class consists in adding to the MHD system penalization terms ensuring that the magnetic field will be solenoidal. The second class consists in formulating the MHD system in term of the vectorial potentiel A that satisfies curl A = B, fulfilling thus automatically the divergence-free constraint. The proposed method is a formulation of the MHD system in term of the mixture of the two former classes. The resulting system is divergence-free constraint preserving and can be approximated by standard Finite Volume methods. Various numerical tests on well-known standard problems in MHD show that this approach is an interesting alternative and opens possibility to use a conservative formulation based on B of the MHD system

A Local Entropy Minimum Principle for Deriving Entropy Preserving Schemes

International audienceThe present work deals with the establishment of stability conditions of finite volume methods to approximate weak solutions of the general Euler equations to simulate compressible flows. In oder to ensure discrete entropy inequalities, we derive a new technique based on a local minimum principle to be satisfied by the specific entropy. Sufficient conditions are exhibited to satisfy the required local minimum entropy principle. Arguing these conditions, a class of entropy preserving schemes is thus derived

Modélisation MHD et Simulation Numérique par Méthode Volumes Finis

International audienceIn the Magneto-HydroDynamic (MHD) equations, the magnetic field has to maintain a constraint of free-divergence. It exists two families of methods dealing with the free-divergence constraint. The first one consist to write the magnetic field as the curl of a potential vector. The second family is the divergence cleaning methods. This presentation proposes a mix of those two methods to keep the free-divergence constraint in the Ideal MHD. The numerical method is based on relaxation schemes. Finally, some numerical tests are performed

Finite volume method in curvilinear coordinates for hyperbolic conservation laws

International audienceThis paper deals with the design of finite volume approximation of hyperbolic conservation laws in curvilinear coordinates. Such coordinates are encountered naturally in many problems as for instance in the analysis of a large number of models coming from magnetic confinement fusion in tokamaks. In this paper we derive a new finite volume method for hyperbolic conservation laws in curvilinear coordinates. The method is first described in a general setting and then is illustrated in 2D polar coordinates. Numerical experiments show its advantages with respect to the use of Cartesian coordinates

Modeling and numerical simulation of transport of charged particles and magnetic field generation in plasma for inertial confinement fusion

Dans le contexte de la production d'énergie de fusion par confinement inertiel, le couplage entre le transport des électrons et la génération des champs magnétiques est un processus crucial. Nous avons développé un nouveau modèle macroscopique capable de prendre en compte ces phénomènes. Ce modèle nous a permis d'analyser l'influence exacte des instabilités de Weibel. A cause de la raideur des équations, nous avons dérivé un nouveau solveur numérique avec lequel des simulations réalistes ont été obtenues et analysées.BORDEAUX1-BU Sciences-Talence (335222101) / SudocSudocFranceF

Modélisation des équations d'Euler bi-températures pour les plasmas de fusion

This work deals with the modeling of fusion plasma by bi-temperature fluid models. First, using non-dimensional scaling of the governing equations, we give the assumptions leading to a bi-temperature model. Then we describe a finite volume method on non-structured meshes to approximate the solutions of this model. The method relies on a relaxation scheme to solve the Riemann problem at the interfaces. The description of the finite volume method that uses the strong conservative form of the equations is given both in Cartesian as well as in cylindrical coordinate systems useful to compute flows inside a torus. Several numerical tests in Cartesian and cylindrical coordinate systems and in different geometries are presented in order to validate the numerical method

Electron temperature anisotropy modeling and its effect on anisotropy-magnetic field coupling in an underdense laser heated plasma

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