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Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis. Numerical analysis naturally finds application in all fields of engineering and the physical sciences, but in the 21st century also the life sciences, social sciences, medicine, business and even the arts have adopted elements of scientific computations. The growth in computing power has revolutionized the use of realistic mathematical models in science and engineering, and subtle numerical analysis is required to implement these detailed models of the world. For example, ordinary differential equations appear in celestial mechanics ; numerical linear algebra is important for data analysis; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology.
Computational physics is the study and implementation of numerical analysis to solve problems in physics for which a quantitative theory already exists. Historically, computational physics was the first application of modern computers in science, and is now a subset of computational science.
Computer science is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. One well known subject classification system for computer science is the ACM Computing Classification System devised by the Association for Computing Machinery.
Theoretical computer science (TCS) is a subset of general computer science that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory.
Computational science, also known as scientific computing or scientific computation (SC), is a rapidly growing field that uses advanced computing capabilities to understand and solve complex problems. It is an area of science which spans many disciplines, but at its core, it involves the development of models and simulations to understand natural systems.
Computational economics is an interdisciplinary research discipline that involves computer science, economics, and management science. This subject encompasses computational modeling of economic systems, whether agent-based, general-equilibrium, macroeconomic, or rational-expectations, computational econometrics and statistics, computational finance, computational tools for the design of automated internet markets, programming tool specifically designed for computational economics and the teaching of computational economics. Some of these areas are unique, while others extend traditional areas of economics by solving problems that are tedious to study without computers and associated numerical methods.
Mesh generation is the practice of creating a mesh, a subdivision of a continuous geometric space into discrete geometric and topological cells. Often these cells form a simplicial complex. Usually the cells partition the geometric input domain. Mesh cells are used as discrete local approximations of the larger domain. Meshes are created by computer algorithms, often with human guidance through a GUI, depending on the complexity of the domain and the type of mesh desired. The goal is to create a mesh that accurately captures the input domain geometry, with high-quality (well-shaped) cells, and without so many cells as to make subsequent calculations intractable. The mesh should also be fine in areas that are important for the subsequent calculations.
Computational mechanics is the discipline concerned with the use of computational methods to study phenomena governed by the principles of mechanics. Before the emergence of computational science as a "third way" besides theoretical and experimental sciences, computational mechanics was widely considered to be a sub-discipline of applied mechanics. It is now considered to be a sub-discipline within computational science.
Engineering mathematics is a branch of applied mathematics concerning mathematical methods and techniques that are typically used in engineering and industry. Along with fields like engineering physics and engineering geology, both of which may belong in the wider category engineering science, engineering mathematics is an interdisciplinary subject motivated by engineers' needs both for practical, theoretical and other considerations outwith their specialization, and to deal with constraints to be effective in their work.
Computational science and engineering (CSE) is a relatively new discipline that deals with the development and application of computational models and simulations, often coupled with high-performance computing, to solve complex physical problems arising in engineering analysis and design as well as natural phenomena. CSE has been described as the "third mode of discovery".
Computational mathematics involves mathematical research in mathematics as well as in areas of science where computing plays a central and essential role, and emphasizes algorithms, numerical methods, and symbolic computations.
David M. Young Jr. was an American mathematician and computer scientist who was one of the pioneers in the field of modern numerical analysis/scientific computing.
The SIAM Journal on Scientific Computing (SISC), formerly SIAM Journal on Scientific & Statistical Computing, is a scientific journal focusing on the research articles on numerical methods and techniques for scientific computation. It is published by the Society for Industrial and Applied Mathematics (SIAM). Jan S. Hesthaven is the current editor-in-chief, assuming the role in January 2016. The impact factor is currently around 2.
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models.
Dimitris Drikakis, PhD, FRAeS, CEng, is a Greek-British applied scientist, engineer and university professor. His research is multidisciplinary. It covers fluid dynamics, computational fluid dynamics, acoustics, heat transfer, computational science from molecular to macro scale, materials, machine learning, and emerging technologies. He has applied his research to diverse fields such as Aerospace & Defence, Biomedical, and Energy and Environment Sectors. He received The William Penney Fellowship Award by the Atomic Weapons Establishment to recognise his contributions to compressible fluid dynamics. He was also the winner of NEF's Innovator of the Year Award by the UK's Institute of Innovation and Knowledge Exchange for a new generation carbon capture nanotechnology that uses carbon nanotubes for filtering out carbon dioxide and other gases.
In numerical analysis, multi-time-step integration, also referred to as multiple-step or asynchronous time integration, is a numerical time-integration method that uses different time-steps or time-integrators for different parts of the problem. There are different approaches to multi-time-step integration. They are based on domain decomposition and can be classified into strong (monolithic) or weak (staggered) schemes. Using different time-steps or time-integrators in the context of a weak algorithm is rather straightforward, because the numerical solvers operate independently. However, this is not the case in a strong algorithm. In the past few years a number of research articles have addressed the development of strong multi-time-step algorithms. In either case, strong or weak, the numerical accuracy and stability needs to be carefully studied. Other approaches to multi-time-step integration in the context of operator splitting methods have also been developed; i.e., multi-rate GARK method and multi-step methods for molecular dynamics simulations.
Ulrich W. Kulisch is a German mathematician specializing in numerical analysis, including the computer implementation of interval arithmetic.
Validated numerics, or rigorous computation, verified computation, reliable computation, numerical verification is numerics including mathematically strict error evaluation, and it is one field of numerical analysis. For computation, interval arithmetic is used, and all results are represented by intervals. Validated numerics were used by Warwick Tucker in order to solve the 14th of Smale's problems, and today it is recognized as a powerful tool for the study of dynamical systems.
Jerzy Respondek is a Polish computer scientist and mathematician, professor at Silesian University of Technology, Gliwice.
