On the numerical resolution of anisotropic equations with high order differential operators arising in plasma physics

In this paper, numerical schemes are introduced for the efficient resolution of anisotropic equations including high order differential operators. The model problem investigated in this paper, though simplified, is representative of the difficulties encountered in the modeling of Tokamak plasmas. The occurrence of high order differential operators introduces specific difficulties for the design of effective numerical methods. On the one hand, regular discretizations of the problem provide matrices characterized by a condition number that blows up with increasing anisotropy strength. On the other hand, matrices issued from Asymptotic-Preserving methods preserve a condition number bounded with respect to the anisotropy strength, nonetheless it scales very poorly as the mesh is refined. Both alternatives reveal to be inoperative in this specific framework to address the targeted values of anisotropy on refined meshes. We therefore introduce two successful methods offering the advantages of each approach: a condition number unrelated to the anisotropy strength and scaling as favorably as standard discretizations with the mesh refinement.