## Asymptotic preserving schemes for linear transport on unstructured meshes

We consider hyperbolic systems with stiff source terms used to approximate the linear transport equation (or Boltzmann linear equation) present in neutronic or radiative transfer. These systems are dependent of a relaxation parameter which generate a diffusion limit. In this work we propose to design finite volumes schemes on unstructured meshes with convergence estimates and stability conditions independent of the relaxation parameter (asymptotic preserving methods [JL90], [JP00], [Jin11], [BT10], [LM09], [Gosse12]). Example of unstructurd mesh: the Kershaw mesh

### Asymptotic preserving schemes for hyperbolic heat equation on unstructured meshes

The classical 2D extension (edge formulation) of Godunov-type asymptotic preserving schemes as Jin-Levermore scheme [JL90] or Gosse-Toscani scheme [GT02] do not converge on unstructured meshes. Indeed the limit diffusion scheme obtained (called "Two Points Flux Approximation" scheme) is consistent only on the meshes which satisfy the Delaunay conditions. Since these geometrical conditions are very restrictives , we propose news convergent methods on a large family of meshes. To design these numerical methods, we propose two ways. The first way is based on alternative formulation ( [Des11], [CDDL09]) of finite volumes methods, called nodal formulation (the numerical fluxes are computed to the nodes contrary to the classical formulation where the fluxes are localized at the middle of the edge) coupled standard methods used to obtain asymptotic preserving schemes (Jin-Levermore method for example). Using two discretizations of the source terms, we obtain two schemes (called JL-(a) et JL-(b)) corresponding to the 2D extension of the Jin-Levermore and Gosse-Toscani schemes. The limit diffusion scheme is convergent with the first order theoretically and the second order numerically under restrictive-less geometrical condition. Furthermore we prove that the JL-(b) is stable and that the semi-implicit time discretization associated to this method is stable under a CFL condition independent of the relaxation parameter.

To illustrate this work we propose to compare the exact solution diffusion (top, left) on Cartesian mesh, the numerical solution given by the classical scheme for hyperbolic heat equation (top, right) on Kershaw mesh (first figure), the numerical solution given by the edge asymptotic preserving scheme (bottom, left) on Kershaw mesh and the numerical solution given by the nodal asymptotic preserving (bottom, right) on Kershaw mesh.

The second way use a different point of view. We propose to design a hyperbolic scheme using a geometrically compatible with a convergent diffusion scheme correctly chooses. After we modify the hyperbolic scheme using the Jin-Levermore method to obtain an asymptotic preserving scheme. In this case we use the Breil-Maire [BM07] and MPFA [AM08] diffusion schemes.

 Key words : Hyperbolic heat equation, unstructured meshes, finite volume methods, nodal scheme, AP schemes, hyperbolic system with source terms.
 Proceedings : An asymptotic preserving scheme for P1 model using classical diffusion schemes on unstructured polygonal meshes, E. Franck, P. Hoch, P. Navaro and G. Samba, ESAIM: PROCEEDINGS, October 2011, Vol. 32, p. 56-75. Proceedings : A priori analysis of asymptotic preserving schemes with the modified equation. B. Després (principal autor), C. Buet et E. Franck. Hyperbolic problems: theory, numerics and applications, AIMS on Applied Mathematics,vol 8, pp 501. Paper : Design of asymptotic preserving finite volume schemes for hyperbolic heat equation on unstructured meshes , C. Buet, B. Després, E. Franck, Numerische Mathematik, October 2012, Volume 122, Issue 2, pp 227-278. Paper : Proof of uniform convergence for a cell-centered AP discretization of the hyperbolic heat equation on general meshes , C. Buet, B. Després, E. Franck, T. Leroy, Mathematics of computation, 12 Septembre 2016.

### Asymptotic preserving schemes for Friedrichs systems and linear transport

The different angular discretizations of the linear transport equation as the discrete ordinate method (Sn models) or the spherical harmonics expansion (Pn models) can be formulate as linear hyperbolic systems with source terms admitting diffusion limit.
Firstly we propose to characterize rigorously the diffusion limit associated to the general linear system with stiff source terms using L2 estimates and Hilbert expansion. This proof is based on an structure assumption for the matrices of the systems.
Secondly we propose asymptotic preserving method for linear general systems with stiff source terms based on algebraic decomposition between a system close to the hyperbolic heat equation and a system which has an insignificant contribution in diffusive regimes. This decomposition gives a very simple asymptotic preserving method using an asymptotic preserving scheme for the hyperbolic heat equation and a classical Rusanov scheme for the second part of the decomposition. This method is close to the micro-macro method ( [LM09], [CL11], [DLNN12]). To finish we verify that the assumptions necessary to use the decomposition are satisfies by the angular discretizations of the transport equation (Pn and Sn models).
 Key words : Friedrichs system, discrete ordinate method, spherical harmonics expansion, unstructured meshes, finite volume methods, nodal scheme, AP schemes, hyperbolic system with source terms, micro-macro decomposition.
 Paper : Asymptotic preserving schemes for Friedrichs systems with stiff relaxation on unstructured meshes: applications to the angular discretization models in linear transport, C. Buet, B. Després, E. Franck, Journal Scientific Computing.

### Asymptotic preserving scheme with maximum principle for non linear hyperbolic model in radiative transfer

In this works we consider a nonlinear system (M1 model) used for radiative transfer simulation ( [Tur05], [Tur12]). This a moment model admitting a diffusion limit, an entropy inequality and a non trivial maximum principle. It is important to preserve these physical properties at the discrete level. However we propose a new method which preserve these properties on unstructured meshes.
We begin by use a analogy between the M1 model and the gas dynamic equation. We obtain a formulation of the M1 model close to the Euler equations which allows to use a Lagrange+remap nodal scheme ( [CDDL09]) coupled the Jin-Levermore method. The scheme is asymptotic preserving with a positive nonlinear diffusion scheme, is entropic and preserve the maximum principle on very distorted meshes. Furthermore we can obtain a semi-implicit time discretization with a CFL condition independent of the relaxation parameter.
 Key words : radiative transfer, unstructured meshes, finite volume methods, non linear moments model, AP schemes, entropy, maximum principle.
 Proceedings : Asymptotic Preserving Finite Volumes Discretization For Non-Linear Moment Model On Unstructured Meshes, C. Buet, B. Després, E. Franck, Finite Volumes for Complex Applications VI Problems and Perspectives, Springer Proceedings in Mathematics Volume 4, 2011, pp 467-474. Paper : An asymptotic preserving scheme with the maximum principle for the M1 model on distorted meshes, C. Buet, B. Després, E. Franck, Comptes Rendus Mathematique, Volume 350, Issues 11-12, June 2012, Pages 633-638.

## Asymptotic preserving and Well-Balanced schemes for non linear hyperbolic systems in fluid mechanics

In this work we study some asymptotic limit the the behavior of the numerical schemes in these regimes. The first point concerns the asymptotic preserving positives schemes (methods with the convergence estimates and stability conditions are independent of the relaxation parameter) and "well-balanced" schemes (methods which preserve the steady states associated to the PDE) for the Euler equations with friction and gravity. The second limit studied is the low mach limit.

### Asymptotic preserving schemes for Euler with friction and gravity

This work consist to apply the previous ideas to nonlinear systems in fluid mechanics, in this case the Euler equations with friction and gravity ( [BMT11], [CCGRS10]). This systems present additional difficulties compared to the equation studied previously. Contrary to the linear transport problems , these systems induce non linear asymptotic diffusion limit, non trivial stationary states and entropy inequalities important to preserve at the discrete level. This work is not finish, however the first results shows that the Lagrange+projection nodal scheme coupled with the Jin-Levermore method gives asymptotic preserving methods with positive second order non linear schemes. Contrary to the Shallow Water equations, the Euler equations admit differential steady states (hydrostatic equilibrium). The scheme allows to preserve the steady states with second order. Modifying the discrete steady states, we can obtain arbitrary high order dsicretization of the steady states.In the futur we propose to study theoreticaly the scheme and the extend the method to others problems.
 Key words : Euler equations, unstructured meshes, finite volume methods, AP schemes, entropy, well-balanced methods, Lagrange+remap scheme.
 Proceedings : Modified Finite Volume Nodal Scheme for Euler Equations with Gravity and Friction, E. Franck, Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects Springer Proceedings in Mathematics & Statistics Volume 77, 2014, pp 285-292.

## Time schemes for the reduced MHD equations

Numerical stability, nonlinear solvers and preconditioning for the reduced MHD equations. The context of this work is the resolution of the MHD equations (Jorek code) and the simulation of the plasma instabilities for the Tokamak as ITER. Firstly we propose to study the theoretical and numerical stability in time for the reduced MHD models. In a second time we study nonlinear time solvers and new preconditioning method for the Jorek code.
For the simulations of edges instabilies in the Tokamak like ITER, we use a code of reduced MHD in the toroidal geometric called JOREK. This code use a implicit discretization with a first order linearization to solve the nonlinear system. In some regime strongly nonlinear, critical numerical instabilies appear. First, we study the the derivation, theoretical and numerical stability of the reduced MHD and bi-fluid models. The first models studied are the reduced models (reduction based on specific forms to the magnetic and velocity fields) associated with the single fluid MHD. For these models we have obtain energy estimates. In the futur we propose to extend these results for the diamagnetic and bi-fluid models in the reduced and full cases. Some others modelizations with other reduction assumptions can be studied. The second part of the works is on the time discretization, linear and nonlinear solvers like Newton methods, continuation methods or adaptive time stepping. The main point is to adapt a new class of preconditioning baed on splitting operators and parabolization of hyperbolic systems on the reduced MHD problems used in JOREK. This method will be coupled with iterative "Jacobian-Free" linear solvers (useful to reduce the memory consumption).
 EUROfusion project research (2014): JOREK, BOUT++ non-linear MHD modelling of MHD instabilities and their control in existing tokamaks and ITER, Becoulet M. (PI), Orain F., Dif-Pradalier G., Latu G., Grandgirard V., Passeron C., Morales J., Nkonga B., Galligo A., Guillard H., Mourrain B., Ratnani A., Futatani S., Ramet P., Lacoste X., Hoelzl M., Sonnendruecker E., Strumberger E., Franck E., Tichmann C., Pamela S., Wilson H., Dudson B., Imada K., Westerhof E., Pavel C., Lessig A.
 EUROfusion Enabling Research Project (2015-2017): Global non-linear MHD modeling in toroidal geometry of disruptions, edge localized modes, and techniques for their mitigation and suppression. Hoelzl M. (PI),Becoulet M., Sonnendruecker E., Strumberger E., Pautasso G., Ratnani A., Orain F., Nardon E., Dif-Pradalier G., Latu G., Grandgirard V., Passeron C., Morales J., Nkonga B., Guillard H., Sangam A., Franck E., Pamela S., Cahyna P., Seidl J., Futatani S., Westerhof E.
 Key words : nonlinear time schemes, preconditioning, stability, multiscale problems, reduced MHD, plasma physics, Jorek code
 Preprint : Energy conservation and numerical stability for the reduced MHD models of the non-linear JOREK code, E. Franck, M. Hölzl, A. Lessig, E. Sonnendrücker.