Strongly anisotropic diffusion problems
Strongly anisotropic diffusion problems occur in various areas of physics such as structural mechanics of plates and shells, geophysical and fast rotating flows, or strongly magnetized plasmas for instance. AP-methods for strongly anisotropic diffusion problems are being developed and applied to plasmas under large magnetic fields.
An example of a strongly magnetized plasma is the ionosphere. The ionospheric plasma is modeled by a transport equation for the plasma density coupled to a strongly anisotropic elliptic equation for the electric potential, with an O(1/eps) coefficient in the direction of the magnetic field. In the so-called 'Striation model', which has been used to produce the video below, the limit eps -> 0 is assumed. This reduces the potential equation to a 2-dimensional one.
The video shows instability patterns which compare well with the observed ‘ionospheric striations’. Striations are ionospheric plasma disturbances which are elongated along the magnetic field line direction. The simulated domain is a magnetic flux tube in which magnetic field –fitted coordinates are used. The upper left insert is a section of the flux tube domain at the level of the equator, while the lower left insert is a section near the ends of the magnetic flux tube. The plasma density is colour-coded. One observes that the disturbance, which is originally located at the equator, instantaneously perturbs the steady ionospheric plasma at the ends of the tube. This propagation of the disturbance is due to the infinite plasma conductivity along the magnetic field lines. This model helps understanding how to prevent Global Positionning Systems against ionospheric perturbations (see the ANR proejct 'IODISSEE').
The assumption that the diffusion coefficient in the magnetic field direction is infinite breaks down at both ends of a magnetic field tube. To improve the results, it is necessary to invert a fully 3-dimensional diffusion problem involving spatial transitions from isotropic (eps = O(1)) to anisotropic (eps << 1) diffusion. A standard scheme leads to the inversion of a highly ill-conditionned matrix. AP schemes lead to matrices with condition number independent of the anisotropy level.
AP schemes based on a 'micro-macro' decompositions have been developed for strongly anisotropic diffusion equations. The picture below displays the solution when the anisotropy direction follows a sinusoid. A caresian mesh has been used. Neither fitted coordinates nor mesh adaptation to the anisotropy direction are required. This feature is a great advantage since fitted coordinates or mesh adaptation can be complex in practice, especially if they must be updated with time in the case of evolution problems.
The pictures below displays the L2 errors produced by this method for anisotropy ratios ranging betwenn 1 and 10-20 for a 100 x 100 mesh (left) and a 200x200 mesh (right). The red curve displays the error of the AP scheme (labelled 'MM') and the green one, the error of a standard method (labelled 'P'). The error of the AP scheme is very small through the whole range of values of the anisotropy ratio, while that of the standard scheme becomes large when the anisotropy ratio becomes small.
In fusion plasmas (such as those prevailing in a Tokamak), the plasma is modeled by the compressible Euler equations with a Lorentz force (here referred to as the 'Euler-Lorentz' model). The large magnetic field limit is combined with a small Mach-number limit and gives rise to a strongly anisotropic elliptic problem for the pressure. AP-methods have been developed for the two-fluid (for electrons and ions) Euler-Lorentz model. In the following picture, the AP-method is compared to a conventional one for a component of the ion momentum (left: Classical scheme ; right: AP -scheme; the momentum is plotted as a function of the two spatial coordinates at a given time). While the former provides the right answer, the latter is unstable. several orders of magnitude in computer efficiency can be gained. Again, unfitted coordinates to the anisotropy geometry are used.
The development of these methods in the framework of the ITER Project is the subject of the ANR project BOOST. Kinetic models, especially the Vlasov equation under a large Lorentz force are also currently investigated.