tokens: the hyperbolic method initially derived for the diffusion equation (Nishikawa 2007 ) to a 2D set of incompressible dissipative MHD equations. This completes the application to advection-diffusion and Navier–Stokes systems. The hyperbolic formulation of the resistive and viscous terms allowed us to construct a simple finite-volume scheme discretization on a rectangular grid, using an approximate Riemann solver to handle the dissipative and advective flux in the same way. For the sake of clarity, a simple second-order upwind flux formulation is chosen in this work. It is shown that our hyperbolic scheme is able to produce stationary solutions for two non-viscous reconnection problems, namely the magnetic annihilation and reconnective diffusion for which exact analytical expressions exist (Watson et al. 1998 ). The first advantage of this approach over traditional schemes is demonstrated through its consistent accuracy over a wide range of magnetic Reynolds numbers. Indeed, the spatial second order of convergence is closely obtained for the main and gradient variables as well. Secondly, the convergence towards the steady state scales only linearly with the cell width h , giving thus a O (1/ h ) acceleration over conventional schemes. Our hyperbolic method has a clear analogy with the hyperbolic divergence cleaning technique used in MHD codes to handle the divergence-free condition of the magnetic field (Dedner et al. 2002 ). Indeed, an additional numerical scalar variable is introduced in order to advect and diffuse the divergence errors to the domain boundaries. A corresponding scalar equation is also added and coupled to the magnetic field induction equation, leading thus to an hyperbolic system. As a major difference, this latter hyperbolic technique is introduced to handle a numerical term and not a physical dissipation term. It is also implemented in the true time evolution of the system contrary to our pseudo-time implementation. The next step in the development of the hyperbolic method for MHD is the extension to time-dependent solutions. Accurate time-dependent computations are possible by using implicit time integration. Indeed, efficient high-order hyperbolic schemes have been proposed for the advection-diffusion equation, with the idea to converge to successive pseudo-steady states that are solutions at successive physical time steps of the time-dependent equations (Mazaheri Nishikawa 2014 ). In order to speed up the convergence procedure iteration at each time step, the scheme presented in this work would probably require some improvements like higher order spatial reconstruction and/or an implicit pseudo-time solver (Nakashima, Watanabe Nishikawa 2016 ). Such time-dependent solutions also require some spatial or/and temporal dependences of the resistivity profile, which are essential in studies related to solar flares. We have checked that our scheme also works when a spatially localized profile is prescribed on the 1D model
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