Dual-field formulation for incompressible MHD equations

  • Palha, Artur (Netherlands eScience Center)

Please login to view abstract download link

This work introduces a dual-field formulation for incompressible MHD systems that satisfies helicity-, mass-, and energy-conservation. The dual-field formulation has been introduced for incompressible Navier-Stokes equations in 2D (using a velocity-vorticity formulation), and in 3D (using a velocity-velocity formulation). This approach exploits the dual nature of physical field quantities by employing two dual sets of equations for the evolution of each field: a primal and a dual equation. Information is exchanged by mixing the two fields in each equation. If this formulation is adequately discretised by a set of discrete function spaces that satisfy a de Rham sequence (e.g., FEEC) it is almost straightforward to construct a discretisation that presents many conservation properties. Moreover, this dual-field approach enables the use of an efficient staggered in time discretisation. This results in a fully quasi-linear time stepping system of equations, circumventing the need to employ expensive iterative solvers for the nonlinear terms. This work expands the dual-field formulation to the incompressible Magneto-Hydrodynamics equations. The resulting formulation employs the same dual velocity-velocity formulation previously presented and couples it to a dual-field formulation of Maxwell's equations. Maxwell's equations explicitly contain this dual-field character, shown by the duality between the electric field E and the electric displacement field D, and the magnetic flux density B and magnetic field strength H. It is shown that this formulation precisely preserves magnetic and cross helicity, the energy law, and the magnetic Gauss law at the discrete level.