https://hal-brgm.archives-ouvertes.fr/hal-03702088Baron, V.V.BaronUB - Université de BordeauxCoudière, Y.Y.CoudièreIMB - Institut de Mathématiques de Bordeaux - Université Bordeaux Segalen - Bordeaux 2 - Université Sciences et Technologies - Bordeaux 1 - UB - Université de Bordeaux - Bordeaux INP - Institut Polytechnique de Bordeaux - CNRS - Centre National de la Recherche ScientifiqueSochala, PierrePierreSochalaBRGM - Bureau de Recherches Géologiques et Minières (BRGM)Adaptive multistep time discretization and linearization based on a posteriori error estimates for the Richards equationHAL CCSD2017A posteriori error estimatesRichards equationDiscrete duality finite volume schemeBackward differentiation formulaAdaptive algorithm[SDU.STU] Sciences of the Universe [physics]/Earth SciencesCOUFFIGNAL, Frédérique2022-06-22 17:22:012022-08-03 04:06:162022-06-22 17:23:20enJournal articles10.1016/j.apnum.2016.10.0051We derive some a posteriori error estimates for the Richards equation. This parabolic equation is nonlinear in space and in time, thus its resolution requires fixed-point iterations within each time step. We measure the approximation error with the dual norm of the residual. A computable upper bound of this error consists of several estimators involving adequate reconstructions based on the degrees of freedom of the scheme. The space and time reconstructions are specified for a two-step backward differentiation formula and a discrete duality finite volume scheme. Our strategy to decrease the computational cost relies on an aggregation of the estimators in three components: space discretization, time discretization, and linearization. We propose an algorithm to stop the fixed-point iterations after the linearization error becomes negligible, and to choose the time step in order to balance the time and space errors. We analyze the influence of the parameters of this algorithm on three test cases and quantify the gain obtained in comparison with a classical simulation.