An integro-differential formulation for magnetic induction in bounded domains: boundary element–finite volume method

Simulations of magnetohydrodynamic (MHD) flows in bounded domains using spectral methods suffer from a number of serious limitations. Alternative methods based on local discretization raise the problem of how to implement non-local boundary conditions for the magnetic field. We have developed a new strategy for the numerical solution of MHD problems in bounded domains, which combines the flexibility of a local discretization with a rigorous formulation of magnetic boundary conditions next to an insulator in arbitrary geometries. In accordance with the character of underlying equations we apply a global integral approach at the boundary and a differential approach inside the conducting domain. The formulation of the boundary problem in terms of primitive variables allows us to combine these approaches and propose a mixed finite volume and boundary element method. We illustrate its efficiency on magnetic diffusion problems in a sphere and in a finite cylinder.