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            <funder>This work was partially supported by National Key R &amp; D Program of the Ministry of Science and Technology of China (Grant NOs. 2020YFA0713400, 2020YFA0713401), National Natural Science Foundation of China (Grant NO. 12471388)</funder>
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              <p>This work introduces a fully discrete scheme for the diffusion-viscous wave equation (DVWe) in the second-order formulation, combining $hp$-DG time-stepping schemes with conforming finite element methods (FEM). Two major theoretical contributions are presented: (1) $hp$-version a priori error estimates in both the energy-norm and DG-norm, which are optimal in the spatial mesh size $h$, temporal step size $\tau$, and temporal polynomial order $q$, yet suboptimal by one order in the spatial polynomial order $p$. Furthermore, for solutions exhibiting weak singularities in time, exponential convergence in terms of the total number of temporal degrees of freedom is proven using the $hp$-refinement strategy. (2) An energy decay estimate that offers explicit bounds involving the diffusive coefficients, discretization parameters, and the Poincar\'{e} inequality constant. A series of numerical experiments are presented to validate the practical performance of the proposed approach.</p>
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