  Charge dynamics play essential role in many practical applications such as
semiconductors, electrochemical devices and transmembrane ion channels. A
Maxwell-Amp\`{e}re Nernst-Planck (MANP) model that describes charge dynamics
via concentrations and the electric displacement is able to take effects beyond
mean-field approximations into account. To obtain physically faithful numerical
solutions, we develop a structure-preserving numerical method for the MANP
model whose solution has several physical properties of importance. By the
Slotboom transform with entropic-mean approximations, a positivity preserving
scheme with Scharfetter-Gummel fluxes is derived for the generalized
Nernst-Planck equations. To deal with the curl-free constraint, the dielectric
displacement from the Maxwell-Amp\`{e}re equation is further updated with a
local relaxation algorithm of linear computational complexity. We prove that
the proposed numerical method unconditionally preserves the mass conservation
and the solution positivity at the discrete level, and satisfies the discrete
energy dissipation law with a time-step restriction. Numerical experiments
verify that our numerical method has expected accuracy and structure-preserving
properties. Applications to ion transport with large convection, arising from
boundary-layer electric field and Born solvation interactions, further
demonstrate that the MANP formulation with the proposed numerical scheme has
attractive performance and can effectively describe charge dynamics with large
convection of high numerical cell P\'{e}clet numbers.
