We deal with the numerical solution of time-dependent partial differential equations using the space-time adaptive discontinuous Galerkin method (STDGM). It is a suitable technique based on a full remeshing of computational grids including anisotropic simplicial elements and varying polynomial approximation degrees. The main idea of the presented approach is to generate a sequence of computational hp-meshes such that the interpolation error is under the given tolerance and the number of degrees of freedom is minimal at each time step. The interpolation error is estimated by a local element-wise higher-order reconstruction. The optimization of the hp-meshes is based on the continuous mesh and error models whose h- variant was developed in [1] and extended to the hp-variant in [2]. The continuous models admit to find semi-optimal grids by means of variation calculus. We present the theoretical background of the proposed method and describe several technical aspects. In comparison to our recent papers [3-4], we introduce several improvements related to the choice of time steps, the choice of the error tolerances, the strategy of the remeshing between time steps, and the recalculation of the solution between two consecutive hp-grids. The numerical experiments demonstrate the accuracy and efficiency of the method, and we also show some applications to compressible flow problems in atmospheric modeling benchmarks from [5]. REFERENCES [1] Adrien Loseille and Frédéric Alauzet, “Continuous mesh framework part I: well-posed continuous interpolation error'', SIAM J. Numer. Anal., Vol. 49, pp.38-60, (2011) [2] V. Dolejší, G. May, and A. Rangarajan, “A continuous hp-mesh model for adaptive discontinuous Galerkin schemes'', Appl. Numer. Math., Vol. 124, pp.1-21, (2018) [3] V. Dolejší, and G. May, “An anisotropic hp-mesh adaptation method for time-dependent problems based on interpolation error control”, J. Sci. Comput. Vol. 95, no. 36, (2023) [4] V. Dolejší, “Non-hydrostatic mesoscale atmospheric modeling by the anisotropic mesh adaptive discontinuous Galerkin method”, Int. J. Comput. Fluid Dyn. (to appear) [5] F. Giraldo and M. Restelli, A study of spectral element and discontinuous Galerkin methods for the Navier–Stokes equations in nonhydrostatic mesoscale atmospheric modeling: Equation sets and test cases”, J. Comput. Phys. 227(8), 3849–3877 (2008)