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Bulletin of the American Mathematical Society

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A review of numerical methods for nonlinear partial differential equations

Author: Eitan Tadmor
Journal: Bull. Amer. Math. Soc. 49 (2012), 507-554
MSC (2010): Primary 35J60, 35K55, 35L65, 35L70, 65Mxx, 65Nxx.
Published electronically: July 20, 2012
MathSciNet review: 2958929
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Abstract: 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.

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Additional Information

Eitan Tadmor
Affiliation: Department of Mathematics and Institute for Physical Science & Technology, Center of Scientific Computation and Mathematical Modeling (CSCAMM), University of Maryland, College Park, Maryland 20742
MR Author ID: 170110
ORCID: 0000-0001-7424-6327

Keywords: Nonlinear PDEs, boundary-value problems, time-dependent problems, well-posed problems, finite-difference methods, finite element methods, finite-volume methods, spectral methods, consistency, accuracy, convergence, stability.
Received by editor(s): March 9, 2011
Received by editor(s) in revised form: May 27, 2012
Published electronically: July 20, 2012
Dedicated: To Heinz-Otto Kreiss with friendship and appreciation
Article copyright: © Copyright 2012 American Mathematical Society