An Asymptotic Preserving method for strongly anisotropic diffusion equations based on field line integration

In magnetized plasma, the magnetic field confines the particles around the field lines. The anisotropy intensity in the viscosity and heat conduction may reach the order of $10^{12}$. When the boundary conditions are periodic or Neumann, the strong diffusion leads to an ill-posed limiting problem. To remove the ill-conditionedness in the highly anisotropic diffusion equations, we introduce a simple but very efficient asymptotic preserving reformulation in this paper. The key idea is that, instead of discretizing the Neumann boundary conditions locally, we replace one of the Neumann boundary condition by the integration of the original problem along the field line, the singular $1/\epsilon$ terms can be replaced by $O(1)$ terms after the integration, so that yields a well-posed problem. Small modifications to the original code are required and no change of coordinates nor mesh adaptation are needed. Uniform convergence with respect to the anisotropy strength $1/\epsilon$ can be observed numerically and the condition number does not scale with the anisotropy.