We review some recent advances in the field of element-based algebraic stabilization for continuous finite element discretizations of nonlinear hyperbolic problems. The main focus is on multidimensional convex limiting techniques designed to constrain antidiffusive element contributions rather than fluxes. We show that the resulting schemes can be interpreted as residual distribution methods. Two kinds of convex limiting can be used to enforce the validity of generalized discrete maximum principles in this context. The first approach has the structure of a localized flux-corrected transport (FCT) algorithm, in which the computation of a low-order predictor is followed by an antidiffusive correction stage. The second option is the use of a monolithic convex limiting (MCL) procedure at the level of spatial semi-discretization. In both cases, inequality constraints are imposed on scalar functions of intermediate states that are required to stay in convex invariant sets.
Distance-Residual Physics-Informed Neural Networks: A Deep Learning Framework for Differential and Partial Differential Inclusions
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine（乌克兰国家科学院B.维尔金低温物理与工程研究所）；Universidad Politécnica de Cartagena（卡塔赫纳理工大学）；Gebze Technical University（格布泽理工大学）；Nicolaus Copernicus University in Toruń（托伦哥白尼大学）
Invariant domain preserving limiting of time explicit and time implicit discretizations for systems of conservation laws
ONERA, Institut Polytechnique de Paris（法国航空航天研究院，巴黎理工学院）；MONHADE, Equipe Inria-ONERA, DAAA, ONERA, Institut Polytechnique de Paris（MONHADE，Inria-ONERA团队，DAAA，法国航空航天研究院，巴黎理工学院）
