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​Key features of the proposed method are accuracy and stability, which will be ensured by the ability of the adaptive technique to preserve the modified mesh as close to the original fixed one as possible. To achieve this goal, a special monitor function is introduced along with an accurate grid reallocation technique. The resulting method, while based on the thermodynamically consistent numerical algorithm for wave and front propagation formulated in terms of excess quantities, incorporates special numerical techniques for an accurate and efficient interface tracking, and a dynamic grid reconstruction function. The numerical results using this method will be compared with results of phase-transition front propagation in solids and densification front propagation in metal foam obtained by applying the fixed-mesh method be used to justify the effectiveness and correctness of the proposed framework. This project will contribute significantly towards the development of corresponding methods in higher dimensions including dynamic crack propagation problems. Development of modern high-resolution finite-volume methods for propagation of discontinuities in solids, as well as of supplementary techniques, is essential for a broad class of problems arising in today's science. The broader impact of this project also includes educational purposes. The method used in this project will be incorporated into future projects for computational mathematics major students who will gain an experience in the state-of-the-art computational science.