A review of numerical methods for nonlinear partial differential equations

By Eitan Tadmor. 2012

207 citations recorded by the source corpus.

. Numerical methods were ﬁrst put into use as an eﬀective tool for solving partial diﬀerential equations (PDEs) by John von Neumann in the mid-1940s. In a 1949 letter von Neumann wrote “the entire computing machine is merely one component of a greater whole, namely, of the unity formed by the computing machine, the mathematical problems that go with it, and the type of planning which is called by both.” The “greater whole” is viewed today as scientiﬁc computation: over the past sixty years, scientiﬁc computation has emerged as the most versatile tool to complement theory and experiments, and numerical methods for solving PDEs are at the heart of many of today’s advanced scientiﬁc computations. Numerical solutions found their way from ﬁnancial models on Wall Street to traﬃc models on Main Street. Here we provide a bird’s eye view on the development of these numerical methods with a particular emphasis on nonlinear PDEs.

Subfields: Computational Mechanics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/04d765c7b4d94064945edc57d6d4ffd7f1d588d0

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Citations in the corpus

Physics-informed machine learning: case studies for weather and climate modelling (2021, background): “Researchers have devoted tremendous effort to developing stable physics-based numerical models that simulate dynamical systems [38].”
Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential Equations (2017, unknown): “We refer, e.g., to Chiaramonte and Kiener (2013), Dehghan et al. (2009), Lagaris et al. (1998), Lee and Kang (1990), Ramuhalli et al. (2005), Tadmor (2012), Thomée (1997), and von Petersdorff and Schwab (2004) for deterministic approximation methods for PDEs, to Bally and Pagès (2003), Bender and…”
On the second-order asymptotical regularization of linear ill-posed inverse problems (2018, unknown): “It is well known that boundedness implies the convergence of consistent schemes for any problem [28], hence, it suffices to show the boundedness of the scheme (45).”
Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks (2019, methodology): “, [30, 94, 95, 97] for deterministic approximation methods for PDEs, cf.”