Duality-based asymptotic-preserving method for highly anisotropic diffusion equations

Abstract

Abstract. The present paper introduces an efficient and accurate numerical scheme for the solution of a highly anisotropic elliptic equation, the anisotropy direction being given by a variable vector field. This scheme is based on an asymptotic preserving reformulation of the original system, permitting an accurate resolution independently of the anisotropy strength and without the need of a mesh adapted to this anisotropy. The counterpart of this original procedure is the larger system size, enlarged by adding auxiliary variables and Lagrange multipliers. This Asymptotic-Preserving method generalizes the method investigated in a previous paper [P. Degond, F. Deluzet, and C. Negulescu, Multiscale Model. Simul., 8(2), 2009/10] to the case of an arbitrary anisotropy direction field.Key words. Anisotropic diffusion, asymptotic preserving scheme, finite element method.AMS subject classifications. 65N30. IntroductionAnisotropic problems are common in the mathematical modeling of physical problems. They occur in various fields of applications such as flows in porous media [3,23] [43] and so on, the list being not exhaustive. The initial motivation for this work is closely related to magnetized plasma simulations such as atmospheric plasma [27,29], internal fusion plasma [4,13] or plasma thrusters [1]. In this context, the medium is structured by the magnetic field, which may be strong in some regions and weak in others. Indeed, the gyration of the charged particles around magnetic field lines dominates the motion in the plane perpendicular to magnetic field. This explains the large number of collisions in the perpendicular plane while the motion along the field lines is rather undisturbed. As a consequence the mobility of particles in different directions differs by many orders of magnitude; this ratio can be as huge as 10 10 . On the other hand, when the magnetic field is weak the anisotropy is much smaller. As the regions with weak and strong magnetic field can coexist in the same computational domain, one needs a numerical scheme which gives accurate results for a large range of anisotropy strengths. The relevant boundary conditions in many fields of application are periodic (for instance in simulations

Hybrid model for the Coupling of an Asymptotic Preserving scheme with the Asymptotic Limit model: The One Dimensional Case

“…The discretized P-model is observed to produce accurate approximations for large and intermediate ε 0 values, but the error rapidly increases for values smaller than 10 −5 . This feature has already been mentioned in precedent works [3,5]. It can be explained by the conditioning of the discretized P-model which blows up with vanishing ε 0 .…”

“…In this paper a strategy is investigated for the spatial coupling of an asymptotic preserving scheme with the asymptotic limit model, associated to a singularly perturbed, highly anisotropic, elliptic problem. This coupling strategy appears to be very advantageous as compared with the numerical discretization of the initial singular perturbation model or the purely asymptotic preserving scheme introduced in previous works [3,5]. The model problem addressed in this paper is well suited for the simulation of a plasma in the presence of a magnetic field, whose intensity may vary considerably within the simulation domain.…”

“…For this reason the discretized system (1) is expected to have a conditioning number that blows up with vanishing values of ε. To circumvent this difficulty an asymptotic preserving scheme has been introduced in [3,5]. The guidelines for the construction of this AP-scheme are briefly sketched here, and we refer to the above articles for the full details.…”

“…The singularly perturbed model problem is stated in section 1 for a simplified one dimensional configuration. An Asymptotic Preserving (AP) reformulation, introduced in a previous work (see [3,5]), is then detailed. This AP-method combines a discretization of the singular perturbation problem for intermediate values of the anisotropy and a discretization of the limit problem for vanishing anisotropic parameter, being thus accurate independently on the intensity of the anisotropy.…”

Numerical study of the plasma tearing instability on the resistive time scale

“…φ ∈ G ψ (with zero Gradient along the field lines), and its fluctuation part φ ∈ A ψ (with zero Average along the field lines). The authors used in former works [10,11] this decomposition for the design of some AP-schemes in the framework of anisotropic elliptic equations.…”