In this work, in order to capture discontinuities correctly in linear elastic solid, augmented internal energy is defined according to the first law of thermodynamics and Hooke’s law. The non-conservative linear elastic system is then rewritten into a conservative form with the help of an augmented total energy equation. We find that the non-physical oscillations occur to the popular HLL and HLLC approximate Riemann solvers when directly applied to simulate the augmented linear elastic solid. We analyze the intrinsic reason by defining a discrepancy factor which can be used to estimate the difference of the total stress across a contact discontinuity, where it is physically required to be continuous. We discover that non-physical oscillations inevitably appear in the vicinity of the contact discontinuity if this factor is away from zero for an approximate Riemann problem solver. In order to overcome this difficulty, we propose an approximate Riemann solver based on the linearized double-shock technique. Theoretical analysis and numerical results show that in comparison to the HLL and HLLC approximate Riemann solvers, the present linearized double-shock Riemann solver can eliminate the non-physical oscillations effectively.