The first four authors have received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska- Curie grant agreement No 777778. David Pardo, Elisabete Alberdi and Judit Muñoz-Matute were partially funded by the Basque Government Consolidated Research Group Grant IT649-13 on “Mathematical Modeling, Simulation, and Industrial Appli- cations (M2SI)” and the Projects of the Spanish Ministry of Economy and Competitiveness with reference MTM2016-76329-R (AEI/FEDER, EU), and MTM2016-81697-ERC/AEI. David Pardo has also received funding from the BCAM “Severo Ochoa” accreditation of excellence SEV-2013-0323 and the Basque Government through the BERC 2014-2017 program. Victor M. Calo was partially funded by the CSIRO Professorial Chair in Computational Geoscience at Curtin University, the Mega-grant of the Russian Federation Government (N 14.Y26.31.0013) and the Deep Earth Imaging Enterprise Future Science Platforms of the Commonwealth Sci- entific Industrial Research Organisation, CSIRO, of Australia. Additional support was provided at Curtin University by The Institute for Geoscience Research (TIGeR) and by the Curtin Institute for Computation. Kristoffer G. van der Zee has received funding from the School of Math- ematical Sciences at University of Nottingham. Judit Muñoz-Matute has received funding from the University of the Basque Country (UPV/EHU) grant No. PIF15/346.

Goal-oriented adaptivity is a powerful tool to accurately approximate physically relevant solution features for partial differential equations. In time dependent problems, we seek to represent the error in the quantity of interest as an integral over the whole space–time domain. A full space–time variational formulation allows such representation. Most authors employ implicit time marching schemes to perform goal-oriented adaptivity as it is known that they can be reinterpreted as Galerkin methods. In this work, we consider variational forms for explicit methods in time. We derive an appropriate error representation and propose a goal-oriented adaptive algorithm in space. For that, we derive the forward Euler method in time employing a discontinuous-in-time Petrov–Galerkin formulation. In terms of time domain adaptivity, we impose the Courant–Friedrichs–Lewy condition to ensure the stability of the method. We provide some numerical results in 1D space + time for the diffusion and advection–diffusion equations to show the performance of the proposed explicit-in-time goal-oriented adaptive algorithm.