Asymptotic-Preserving methods

The numerical resolution of highly anisotropic physical problems poses considerable numerical challenges, in particular it is difficult to capture the behaviour of physical phenomena characterized by anisotropic features with affordably low numerical costs. In the class of problems addressed in our work the anisotropy is aligned with a vector field which may be variable in space/time. The order of anisotrpy is O(1/eps), where eps is a positive and small parameter. Such problems are encountered in many physical applications, for example flows in porous media, semiconductor modeling, quasineutral plasma simulations, the list of possible applications being not exhaustive. The motivation of this work is closely related to the magnetized plasma simulations such as atmospheric plasmas, internal fusion plasmas or plasma thrusters. In this case the anisotropy direction is defined by a magnetic field confining the particles around the field lines, the anisotropy parameter eps reaching orders of magnitude as low as 10^-10.

The difficulty with these anisotropic problems is that the standard methods become very badly conditioned where eps approaches 0. Moreover, putting eps=0 in the initial equation leads to a singular problem. The philosophy of the Asymptotic-Preservong (AP) schemes resides in a suitable reformulation such that conditioning is no longer eps dependant and the correct limit problem is recovered in the limit of infinite anisotropy. We have developped several AP schemes in the context of elliptic equations.

• Duality Based (DB) scheme: the solution u is decomposed into two orthogonal parts: u=p+q, with p constant in the direction of anisotropy and the q of average zero in the direction of anisotropy. Ensuring those properties requires introduction of three additional Lagrange multipliers resulting in a system of 5 equations
• Micro-Macro (MM) scheme: the solution u is decomposed into u = p + eps q, with p constant in the direction of anisotropy and q a complement belonging to a suitable choosen and easy to discretize space, carying informations about fluctuations, but not of average zero. The resulting system has 2 equations
• Micro-Macro with Stabilization: similar decomposition as in MM, but now p and q are orthogonal, orthogonality provided by a penalty stabilization term inspired by stabilization methods for Stokes problem. Contrary to DB and MM, the MMS scheme allows for anisotropy following closed lines
• Two Field Iterated (TFI) scheme: a solution to the initial higly anisotropic equation is obtained by iterative resolution of two only mildy anisotropic equations for two alternating fields. As general as the MMS but significantly improves conditioning, faster resolution time for fine meshes than MM and MMS despite of iterative process.

All methods give comparable precission. The pictures below display the L2 (left) and H1 (right) error produced on the mesh 200x200 with Q2 finite elements for values of eps ranging from 10^-20 to 10 (horizontal axis). Three methods are compared : the AP scheme, a standard discretization of the problem (P) and the limit model (L). The AP scheme is precise in the whole range of the anisotropy strength while the P model breaks for small values of eps and the L model is incorrect when anisotropy strength is close to 1.

Both methods seems to be robust and work well even for highly oscillating anisotropy field. Figures below show L2 and H1 errors for anisotropy ratios equal to 1 (left) and 10^-20 (right). In the presented test case the anisotropy follows a sinusoidal shape with m periods. Errors are plotted as a function of m ranging from 1 to 50. The mesh size is 400x400 and Q2 finite elements are used. It is remarkable that the relative error in both norms is less than 1% as long as more than only 20 mesh points per period are used.

The MM scheme can be easily generalized to the case of nonlinear anisotropic hyperbolic equations, where nonlinearity appears only in one selected direction. The first order scheme based on implicit Euler time discretization is straight forward. The second order scheme however demands more attention. Standard Crank-Nicolson method does not appear to be Asymptotic Preserving giving the oscilating numerical solutions for big times steps. In order to achieve a good precision and convergence one has to choose a time step of the order of the anisotropy strength. This is not the case for implicit Euler method. If a second order time convergence is desired one has to apply a method which is L-stable, such as a two step Diagonally Implicit Rung-Kutta (DIRK) scheme. The time convergence of AP Euler (E-MM), AP DIRK (RK-MM) and the standard (P) schemes for eps = 10^-10 is shown on Figure below.

Note that the standard method fails to converge, E-MM is, as predicted, of the first order and RK-MM is of the second order for large time steps. For small time steps the error due to the time discretization becomes comparable with space discratization error thus the convergence is lost for RK-MM scheme.

Publications

• C. Yang, F. Deluzet, J. Narski, On the numerical resolution of anisotropic equations with high order differential operators: application to tokamak plasma physics,J. Comput. Phys. (2019), Vol. 386, 502-523
• F. Deluzet, J. Narski, A two field iterated Asymptotic-Preserving method for highly anisotropic elliptic equations, Multiscale Model. Simul. 17(1) (Jan. 2019) 434–459
• A. Lozinski, J. Narski, C. Negulescu, Numerical analysis of an asymptotic-preserving scheme for anisotropic elliptic equations, submitted, preprint: arXiv:1507.00879
• J. Narski, M. Ottaviani, Asymptotic Preserving scheme for strongly anisotropic parabolic equations for arbitrary anisotropy direction, Computer Physics Communications 185 (2014) pp. 3189-3203, preprint: arXiv:1303.5219
• A. Lozinski, J. Narski, C. Negulescu, Highly anisotropic temperature balance equation and its numerical solution using asymptotic-preserving schemes of second order in time, M2AN 48 (2014) 1701–1724, preprint: arXiv:1203.6739
• P. Degond, A. Lozinski, J. Narski & C. Negulescu, An Asymptotic-Preserving method for highly anisotropic elliptic equations based on a micro-macro decomposition, J. Comput. Phys. (2012), Vol. 231(7), 2724-2740, preprint: http://arxiv.org/abs/1102.0904
• P. Degond, F. Deluzet, A. Lozinski, J. Narski & C. Negulescu, Duality-based Asymptotic-Preserving method for highly anisotropic diffusion equations, Commun. Math. Sci. (2012), Vol. 10(1), 1-31, preprint: http://arxiv.org/abs/1008.3405
• P. Degond, F. Deluzet, D. Maldarella, J. Narski, C. Negulescu, M. Parisot, Hybrid model for the Coupling of an Asymptotic Preserving scheme with the Asymptotic Limit model: The One Dimensional Case, ESAIM: Proc. 32, 23-30 (2011)