A numerical algorithm for uni-axial inelastic wave propagation due to impact-type loading is presented. The algorithm is based on the linear-elastic multiple-field formulation, in which inelastic parts of strain are considered as eigenstrains acting upon a linear-elastic background structure. The solution thus can be consistently found by superposition of altogether elastic waves, where the fictitious eigenstrains are calculated from the inelastic constitutive equations by numerical integration. Due to the physical background of the method, the procedure turns out to be both, computationally accurate and numerically stable. In the present contribution, the algorithm is realized in C++ and applied to the loading-unloading problem of a semi-infinite rod of Maxwell material. The accuracy of the algorithm is demonstrated by comparing to an analytic solution which is derived in the form suitable for comparison.