Abstract: In layered media, the solution of Maxwell’s equations suffers a strong or weak discontinuity at the layer boundaries. Finite-difference schemes providing convergence on strong discontinuities have been proposed for the first time. These are conservative bicompact two-point schemes with mesh nodes lying on the layer boundaries. A fundamentally new technique for taking into account the medium dispersion is proposed. All these features ensure the second order of accuracy of the schemes on discontinuous solutions. Numerical examples illustrating these results are given. © 2020, Pleiades Publishing, Ltd.