We develop a new unified high-order method for continuum simulations of flows in gases, liquids and elastic-plastic solids. We show that a high order (10th order) compact finite difference scheme in conjunction with the Localized Artificial Diffusivity (LAD) method for numerical regularization is suitable for problems involving high deformation flows in elastic-plastic solids. The LAD method has previously been used successfully for simulations of fluid flows. This choice of numerical algorithms allows for coupled fluid and solid flows that are of significant engineering interest. Particular emphasis is laid on the choice of the artificial diffusivity parameters in order to sufficiently capture discontinuities in all the aforementioned continuum media with minimal added dissipation. Test cases in one and two dimensions are shown to demonstrate the feasibility and accuracy of the proposed approach. © 2016, American Institute of Aeronautics and Astronautics Inc, AIAA. All rights reserved.