I will briefly survey some applied mathematics problems that are motivated by magnetized plasma problems (including gyrokinetic simulations of turbulence in fusion energy devices) but have broader interest as well. (1) There is a massive literature on advection algorithms for various types of problems (compressible flows with shocks, incompressible flows, Hamiltonian systems) and I will discuss some of the tradeoffs and recent advances for a paradigm problem of 2D incompressible vorticity advection. (2) Magnetized plasmas typically have very fast diffusion of particles along magnetic field lines, and much slower diffusion across field lines. This type of anisotropic diffusion occurs in a wide range of applications (including geology, diffusion-tensor Magnetic Resonance Imaging of biological systems, and image filtering), but it can be surprisingly difficult to maintain positivity of the diffusing quantity with standard algorithms. (3) Symplectic integrators (which preserve certain conservation properties of the underlying equations) often involve implicit methods, which will usually require iterative methods to solve them. We show one way of doing this which results in each iteration being symplectic (not just in the limit that the iterations converge).
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