Over the last 15 years, literature on nondestructive testing has shown that the generation of higher harmonics and nonlinear mixing of waves could be used to obtain the nonlinearity parameters of an elastic medium and thereby gather information about its state, e.g., aging and fatigue. To design ultrasound measurement setups based on these phenomena, efficient numerical modeling tools are needed. In this paper, the iterative nonlinear contrast source method for numerical modeling of nonlinear acoustic waves is extended to the one-dimensional elastic case. In particular, nonlinear mixing of two collinear bulk waves (one compressional, one shear) in a homogeneous, isotropic medium is considered, taking into account its third-order elastic constants ( A, B, and C). The obtained results for nonlinear propagation are in good agreement with a benchmark solution based on the modified Burgers equation. The results for the resonant waves that are caused by the one-way and two-way mixing of primary waves are in quantitative agreement with the results in the literature [Chen, Tang, Zhao, Jacobs, and Qu, J. Acoust. Soc. Am. 136(5), 2389-2404 (2014)]. The contrast source approach allows the identification of the propagating and evanescent components of the scattered wavefield in the wavenumber-frequency domain, which provides physical insight into the mixing process and explains the propagation direction of the resonant wave.