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A high-order implicit finite element method for integrating the two-fluid magnetohydrodynamic equations in two dimensions. (English) Zbl 1388.76442
Summary: We describe a new method for solving the time-dependent two-fluid magnetohydrodynamic (2F-MHD) equations in two dimensions that has significant advantages over other methods. The stream-function/potential representation of the velocity and magnetic field vectors, while fully general, allows accurate description of nearly incompressible fluid motions and manifestly satisfies the divergence condition on the magnetic field. Through analytic manipulation, the split semi-implicit method breaks the full matrix time advance into four sequential time advances, each involving smaller matrices. The use of a high-order triangular element with continuous first derivatives ($$C^{1}$$ continuity) allows the Galerkin method to be applied without introduction of new auxiliary variables (such as the vorticity or the current density). These features, along with the manifestly compact nature of the fully node-based $$C^{1}$$ finite elements, lead to minimum size matrices for an unconditionally stable method with order of accuracy $$h^{4}$$. The resulting matrices are compatible with direct factorization using SuperLU_dist. We demonstrate the accuracy of the method by presenting examples of two-fluid linear wave propagation, two-fluid linear eigenmodes of a tilting cylinder, and of a challenging nonlinear problem in two-fluid magnetic reconnection.

##### MSC:
 76W05 Magnetohydrodynamics and electrohydrodynamics 76M10 Finite element methods applied to problems in fluid mechanics
##### Software:
M3D-C; SEL/HiFi; SuperLU
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##### References:
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