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The force‐free (or low inertia) limit of magnetohydrodynamics (MHD) can be applied to many astrophysical objects, including black holes, neutron stars and accretion discs, where the electromagnetic field is so strong that the inertia and pressure of the plasma can be ignored. This is difficult to achieve with the standard MHD numerical methods because they still have to deal with plasma inertial terms even when these terms are much smaller than the electromagnetic terms. Under the force‐free approximation, the plasma dynamics is entirely determined by the magnetic field. The plasma provides the currents and charge densities required by the dynamics of electromagnetic fields, but these currents carry no inertia. We present a high‐order Godunov scheme to study such force‐free electrodynamics. We have implemented weighted essentially non‐oscillatory (WENO) spatial interpolations in our scheme. An exact Riemann solver is implemented, which requires spectral decomposition into characteristic waves. We advance the magnetic field with the constrained transport (CT) scheme to preserve the divergence‐free condition to machine round‐off error. We apply the third‐order total variation diminishing (TVD) Runge–Kutta scheme for the temporal integration. The mapping from face‐centred variables to volume‐centred variables is carefully considered. Extensive testing are performed to demonstrate the ability of our scheme to address force‐free electrodynamics correctly. We finally apply the scheme to study relativistic magnetically dominated tearing instabilities and neutron star magnetospheres.

A high‐order WENO‐based staggered Godunov‐type scheme with constrained transport for force‐free electrodynamics