The Cartesian grid discontinuous Galerkin (cgDG) [1] method offers a powerful immersed boundary framework for solving hyperbolic partial differential equations, combining the flexibility of Cartesian meshes with high-order accuracy. However, for transient phenomena involving shock waves or geometric singularities—such as those found in electromagnetic scattering—static uniform discretizations become computationally inefficient. This work presents a fully automatic hp-adaptive strategy designed to optimize the cgDG spatial discretization dynamically within the time integration loop. We propose and compare two distinct local error measurement techniques to guide the adaptation process: an explicit error indicator derived from the residual of the weak formulation, and a multigrid-based estimator that utilizes a hierarchy of coarser and finer auxiliary meshes to predict error distribution and solution regularity. These indicators are coupled with specific criteria to automatically discern between element subdivision (h-refinement) and polynomial degree enrichment (p-refinement). Furthermore, to ensure accuracy is maintained during mesh transitions, we introduce a rigorous projection operator based on the DG formulation for transferring fields between evolving discretizations. Numerical benchmarks are presented to validate the methodology and compare the stability and robustness of both proposed strategies, proving them to be a reliable tool for minimizing computational cost while maintaining high-fidelity results in transient electromagnetic simulations. ACKNOWLEDGEMENTS The authors gratefully acknowledge the financial support of: Grant PID2022-141512NB-I00 funded by MCIN/AEI/10.13039/501100011033 and ERDF/EU. Grant FPU17/03993 funded by “Ministerio de Universidades”. REFERENCES [1] Navarro García, H. (2025). High-order techniques in the Cartesian grid finite element method based on the discontinuous Galerkin formulation. Universidad Politécnica de Valencia. https://riunet.upv.es/handle/10251/229753