We propose a novel finite-difference time-domain (FDTD) scheme for the solution of the Maxwell's equations in which linear dispersive effects are present. The method uses high-order accurate approximations in space and time for the dispersive Maxwell's equations written as a second-order vector wave equation with a time-history convolution term. The modified equation approach is combined with the recursive convolution (RC) method to develop high-order approximations accurate to any desired order in space and time. High-order-accurate centered approximations of the physical Maxwell interface conditions are derived for the dispersive setting in order to fully restore accuracy at discontinuous material interfaces. Second-and fourth-order accurate versions of the scheme are presented and implemented in two spatial dimensions for the case of the Drude linear dispersion model. The stability of these schemes is analyzed. Finally, our approach is also amenable to curvilinear numerical grids if used with appropriate generalized Laplace operator. AMS subject classifications. 65M06, 78M20, 78A40, 35L05, 35Q60, 35Q61, 65Z05, 65M12 1. Introduction. In many materials, such as metal or biological tissue, lossy and dispersive effects resulting from the electronic response of the material have a considerable effect on electromagnetic wave propagation and must be included in the model. These materials are referred to as "dispersive" media. Simple models of dispersive optical media often use a classical linear Newtonian description to describe the response of charge carriers to an external electric field. Three common examples are the Debye, Lorentz, and Drude models, all of which can be generalized using the Padé approximant method [36, 37]. Accurate time-domain simulation of dispersive optical effects is important for various applications in optical computing, imaging, and sensing, where transient electromagnetic waves play an important role in the system of interest. Furthermore, these applications often involve multiple materials that meet along a common interface, which results in a situation with discontinuous material parameters and the possibility of important interface phenomenon. For example, the propagating surface waves ("surface plasmon polaritons") at interfaces between dielectrics and metals, made possible by the dispersive nature of the metal, are of considerable interest in the field of plasmonics, and dispersion plays an important role in their excitation and attenuation [26, 34]. It is therefore desirable to derive high-order accurate time-domain solvers to simulate dispersive electromagnetic wave propagation and have the capability to treat material interfaces. Perhaps the most common time-domain scheme is the ubiquitous second-order accurate Yee scheme [42, 45]. This finite-difference time-domain (FDTD) solver for computational electromagnetics solves the first-order form of Maxwell's equations on a grid which is staggered in time and space. The scheme is attractive for many reasons including its efficiency an...