A Finite-Volume Algorithm for Three-Dimensional Magnetohydrodynamics on an Unstructured, Adaptive Grid in Axially Symmetric Geometry

A new finite-volume algorithm for the solution of the time-dependent, nonideal magnetohydrodynamic (MHD) equations in cylindrical (r, ?,z) geometry is presented. The boundary geometry is assumed to be axially symmetric, but it can have arbitrary shape and connectivity in the poloidal (r,z) plane. The dynamics of the fluid is fully three-dimensional. A two-dimensional, unstructured, adaptive grid of triangles is used to describe the poloidal geometry. A pseudospectral algorithm with fast Fourier transforms is used for the periodic toroidal (?) direction. The grid can be dynamically refined or coarsened by adding or deleting points to adapt to evolving fine-scale structures in the solution. The algorithm exactly conserves total mass, momentum, energy, and magnetic flux, and identically preserves the solenoidal properties of the magnetic field and the current density. Examples of the application of the algorithm to two-dimensional hydrodynamic and MHD shocks, the linear growth of a resistive tearing instability in a tokamak, and the linear growth and nonlinear saturation of three-dimensional kink instabilities in toroidal geometry are given.

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