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Uniform convergent scheme for strongly anisotropic diffusion equations with closed field lines

In magnetized plasma, the magnetic field confines particles around field lines. The ratio between the intensity of the parallel and perpendicular viscosity or heat conduction may reach the order of $10^{12}$. When the magnetic fields have closed field lines and form a "magnetic island", the convergence order of most known schemes depends on the anisotropy strength. In this paper, by integration of the original differential equation along each closed field line, we introduce a simple but very efficient asymptotic preserving reformulation, which yields uniform convergence with respect to the anisotropy strength. Only slight modification to the original code is required and neither change of coordinates nor mesh adaptation is needed. Numerical examples demonstrating the performance of the new scheme are presented.