Asymptotic-preserving methods for an anisotropic model of electrical potential in a tokamak

A 2D nonlinear model for the electrical potential in the edge plasma in a tokamak generates a stiff problem due to the low resistivity in the direction parallel to the magnetic field lines. An asymptotic-preserving method based on a micro-macro decomposition is studied in order to have a well-posed problem, even when the parallel resistivity goes to $0$. Numerical tests with a finite difference scheme show a bounded condition number for the linearised discrete problem solved at each time step, which confirms the theoretical analysis on the continuous problem.Comment: 8 page

An optimal penalty method for a hyperbolic system modeling the edge plasma transport in a tokamak

The penalization method is used to take account of obstacles, such as the limiter, in a tokamak. Because of the magnetic confinement of the plasma in a tokamak, the transport occurs essentially in the direction parallel to the magnetic field lines. We study a 1D nonlinear hyperbolic system as a simplified model of the plasma transport in the area close to the wall. A penalization which cuts the flux term of the momentum is studied. We show numerically that this penalization creates a Dirac measure at the plasma-limiter interface which prevents us from defining the transport term in the usual distribution sense. Hence, a new penalty method is proposed for this hyperbolic system. For this penalty method, an asymptotic expansion and numerical tests give an optimal rate of convergence without spurious boundary layer. Another two-fields penalization has also been implemented and the numerical convergence analysis when the penalization parameter tends to $0$ reveals the presence of a boundary layer

On the well-posed coupling between free fluid and porous viscous flows

International audienceWe present a well-posed model for the Stokes/Brinkman problem with {\em jump embedded boundary conditions (J.E.B.C.)} on an immersed interface. It is issued from a general framework recently proposed for fictitious domain problems. Our model is based on algebraic transmission conditions combining the stress and velocity jumps on the interface $\S$ separating the fluid and porous domains. These conditions are well chosen to get the coercivity of the operator. Then, the general framework allows to prove the global solvability of some models with physically relevant stress or velocity jump boundary conditions for the momentum transport at a fluid-porous interface. The Stokes/Brinkman problem with {\em Ochoa-Tapia \& Whitaker (1995)} interface conditions and the Stokes/Darcy problem with {\em Beavers \& Joseph (1967)} conditions are both proved to be well-posed by an asymptotic analysis. Up to now, only the Stokes/Darcy problem with {\em Saffman (1971)} approximate interface conditions was known to be well-posed