Uniform Convergent Schemes For Strongly Anisotropic Diffusion Equations

Anisotropic diffusion is encountered in many physical applications,including flows in porous media,heat conduction in fusion plasmas,atmospheric or oceanic flows and so on.Strong anisotropic diffusion equation is studied in this paper with Neumann boundary conditions,closed field line,discontinuous,vanishing diffusivity.Our goal is to establish uniform convergent scheme for these systems.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 1012.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/? terms can be replaced by O(1)terms after the integration,so that yields a well-posed problem.To discretize the magnetic fields that have closed field lines,field-aligned grids induce an ill-posed system in the strongly anisotropic limit,while the convergence or-ders of most known schemes with non-aligned grids depend on the anisotropy strength.This paper introduces a simple but very efficient asymptotic preserving reformulation for the magnetic field that has closed field lines.The new reformulated system removes the ill-posedness in the strongly anisotropic limit.The key idea is that we cut each of the closed field line at some point(x0,y0)and replace the local discretization of the equation at the point(x0,y0)by some average of the differential equation along the field line.The singular 1/? terms can be replaced by O(1)terms after the integration,so that yields a well-posed problem.For the strongly anisotropic diffusion equation considered in this paper,the AP methods is required.The novelties of our scheme are listed below:1)Small modifica-tions to the original code are required,which makes it attractable to engineers.The idea can be coupled with most standard discretizations and the computational cost keeps almost the same.2)Extensions to space dependent ?,? and ? are straight for-ward.Uniform convergence with respect to the anisotropy strength ? can be observed numerically and the condition number does not scale with the anisotropy.3)No change of coordinates nor mesh adaptation are required.The computational cost of magnetic field aligned coordinate is expensive which motivates the use of coordinates and meshes that are independent of the anisotropic direction.We propose two Tailored Finite Point methods(TFPM)for the discontinuous,vanishing diffusivity,strongly anisotropic diffusion equation.The diffusion coefficient can be very small in one direction in some part of the domain and be discontinuous at the interfaces.Uniform convergence can be observed numerically in the vanish-ing diffusivity limit,even when the solution exhibits interface and boundary layers.Moreover,when the diffusivity is along the coordinates,the positivity and maximum principle can be proved.We use the value as well as their derivatives at the grid points to construct the scheme,which makes it can achieve good accuracy and convergence for the derivatives as well,even when there exhibit boundary or interface layers.