ALADINS: an ALgebraic splitting time ADaptive solver for the Incompressible Navier-Stokes equations. We address a time-adaptive solver specifically devised for the incompressible Navier-Stokes (INS) equations. One of the challenging issues in this context is the identification of a reliable a posteriori error estimator. Typical strategies are based on the combination of the solutions computed either with two different time steps or two schemes with different accuracy. In this paper, we move from the pressure correction algebraic factorizations formerly proposed by F. Saleri and A. Veneziani [SIAM J. Numer. Anal. 43, No. 1, 174–194 (2005; Zbl 1128.76047)]. These schemes feature an intrinsic hierarchical nature, such that an accurate solution for the pressure is obtained by computing intermediate low-order guesses. The difference between the two estimates provide a natural a posteriori estimator. After introducing the incremental formulation of the pressure correction schemes, we address the properties of this approach, including extensive implementation details. Numerical results presented refer to 2D and 3D unstructured problems, with a particular emphasis on cardiovascular problems, which are expected to largely benefit from time-adaptive solvers. In memory of F. Saleri (1965–2007).

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  1. DeCaria, Victor; Gottlieb, Sigal; Grant, Zachary J.; Layton, William J.: A general linear method approach to the design and optimization of efficient, accurate, and easily implemented time-stepping methods in CFD (2022)
  2. Layton, William; McLaughlin, Michael: Doubly-adaptive artificial compression methods for incompressible flow (2020)
  3. Viguerie, Alex; Veneziani, Alessandro: Algebraic splitting methods for the steady incompressible Navier-Stokes equations at moderate Reynolds numbers (2018)
  4. Baroli, Davide; Cova, Cristina Maria; Perotto, Simona; Sala, Lorenzo; Veneziani, Alessandro: Hi-POD solution of parametrized fluid dynamics problems: preliminary results (2017)
  5. Quarteroni, A.; Manzoni, A.; Vergara, C.: The cardiovascular system: mathematical modelling, numerical algorithms and clinical applications (2017)
  6. Rebholz, Leo G.; Xiao, Mengying: Improved accuracy in algebraic splitting methods for Navier-Stokes equations (2017)
  7. Gallo, Diego; Lefieux, Adrien; Morganti, Simone; Veneziani, Alessandro; Reali, Alessandro; Auricchio, Ferdinando; Conti, Michele; Morbiducci, Umberto: A patient-specific follow up study of the impact of thoracic endovascular repair (TEVAR) on aortic anatomy and on post-operative hemodynamics. (2016)
  8. Lefieux, A.; Auricchio, F.; Conti, M.; Morganti, S.; Reali, A.; Trimarchi, S.; Veneziani, A.: Computational study of aortic hemodynamics: from simplified to patient-specific geometries (2016)
  9. Deparis, Simone; Grandperrin, Gwenol; Quarteroni, Alfio: Parallel preconditioners for the unsteady Navier-Stokes equations and applications to hemodynamics simulations (2014)
  10. Turek, Stefan; Mierka, Otto; Hysing, Shuren; Kuzmin, Dmitri: Numerical study of a high order 3D FEM-level set approach for immiscible flow simulation (2013)
  11. Veneziani, Alessandro; Villa, Umberto: ALADINS: an ALgebraic splitting time ADaptive solver for the Incompressible Navier-Stokes equations (2013)