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oai:recercat.cat:2117/1042342026-01-21T06:56:38Zcom_2072_1033col_2072_452950

Differentiable monotonicity-preserving schemes for discontinuous Galerkin methods on arbitrary meshes Badia, Santiago Bonilla de Toro, Jesús Hierro Fabregat, Alba Universitat Politècnica de Catalunya. Departament d'Enginyeria Civil i Ambiental Centre Internacional de Mètodes Numèrics en Enginyeria Universitat Politècnica de Catalunya. ANiComp - Anàlisi numèrica i computació científica Àrees temàtiques de la UPC::Matemàtiques i estadística::Anàlisi numèrica Galerkin methods Finite elements Discrete maximum principle Monotonicity Shock capturing Discontinuous Galerkin Local extrema diminishing Galerkin, Mètodes de This work is devoted to the design of interior penalty discontinuous Galerkin (dG) schemes that preserve maximum principles at the discrete level for the steady transport and convection–diffusion problems and the respective transient problems with implicit time integration. Monotonic schemes that combine explicit time stepping with dG space discretization are very common, but the design of such schemes for implicit time stepping is rare, and it had only been attained so far for 1D problems. The proposed scheme is based on a piecewise linear dG discretization supplemented with an artificial diffusion that linearly depends on a shock detector that identifies the troublesome areas. In order to define the new shock detector, we have introduced the concept of discrete local extrema. The diffusion operator is a graph-Laplacian, instead of the more common finite element discretization of the Laplacian operator, which is essential to keep monotonicity on general meshes and in multi-dimension. The resulting nonlinear stabilization is non-smooth and nonlinear solvers can fail to converge. As a result, we propose a smoothed (twice differentiable) version of the nonlinear stabilization, which allows us to use Newton with line search nonlinear solvers and dramatically improve nonlinear convergence. A theoretical numerical analysis of the proposed schemes shows that they satisfy the desired monotonicity properties. Further, the resulting operator is Lipschitz continuous and there exists at least one solution of the discrete problem, even in the non-smooth version. We provide a set of numerical results to support our findings. Peer Reviewed Postprint (author's final draft) 2017-06 Article Badia, S., Bonilla, J., Hierro, A. Differentiable monotonicity-preserving schemes for discontinuous Galerkin methods on arbitrary meshes. "Computer methods in applied mechanics and engineering", Juny 2017, vol. 320, p. 582-605. 0045-7825 https://hdl.handle.net/2117/104234 10.1016/j.cma.2017.03.032 eng http://www.sciencedirect.com/science/article/pii/S0045782516319090 http://creativecommons.org/licenses/by-nc-nd/3.0/es/ Open Access Attribution-NonCommercial-NoDerivs 3.0 Spain 24 p. application/pdf