A non-staggered, conservative, image, finite-volume scheme for 3D implicit extended magnetohydrodynamics in curvilinear geometries Original Research Article

In the development of spatial difference schemes for magnetohydrodynamics (MHD), the preservation of continuum properties such as conservation of mass, momentum, and energy, as well as required electromagnetic constraints (image, where image is the electrical current), is desirable to preserve numerical accuracy. Moreover, simplicity of the scheme is also a desirable feature, particularly when an implicit implementation is considered (the focus of this paper). We propose here a finite-volume, cell-centered (non-staggered) scheme for the extended MHD formulation that: (1) is suitable for implicit implementations in arbitrary curvilinear geometries, (2) is conservative, (3) preserves both the magnetic field and the electrical current solenoidal to machine precision, and (4) is linearly and nonlinearly stable in the absence of numerical and physical dissipation. Crucial to the viability of the scheme is the use of a clever interpolation scheme (ZIP [Hirt, J. Comput. Phys. 2 (1968) 339–355]), the proper treatment of boundary conditions in curvilinear geometry, and a novel treatment of geometric source terms in the momentum equation that ensures their exact cancellation in the absence of pressure forces.