Computational science (or scientific computing) is the field of study concerned with constructing mathematical models and numerical solution techniques and using computers to analyse and solve scientific, social scientific and engineering problems. In practical use, it is typically the application of computer simulation and other forms of computation to problems in various scientific disciplines. A mathematical model uses mathematical language to describe a system. ... For the scientific journal named Science, see Science (journal). ... The social sciences are groups of academic disciplines that study the human aspects of the world. ... Engineering is the discipline and profession of applying scientific knowledge and utilizing natural laws and physical resources in order to design and implement materials, structures, machines, devices, systems, and processes that realize a desired objective and meet specified criteria. ... This article is about computer modeling within a scientific medium. ... Look up computation in Wiktionary, the free dictionary. ...

The field is distinct from computer science (the mathematical study of computation, computers and information processing). It is also different from theory and experiment which are the traditional forms of science and engineering. The scientific computing approach is to gain understanding, mainly through the analysis of mathematical models implemented on computers. Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ... Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. ... Look up computation in Wiktionary, the free dictionary. ... This article is about the machine. ... In general, information processing is the changing (processing) of information in any manner detectable by an observer. ... This article is about the machine. ...

Scientists and engineers develop computer programs, application software, that model systems being studied and run these programs with various sets of input parameters. Typically, these models require massive amounts of calculations (usually floating-point) and are often executed on supercomputers or distributed computing platforms. A computer program (often simply called a program) is an example of computer software that prescribes the actions (computations) that are to be carried out by a computer. ... Application software is a subclass of computer software that employs the capabilities of a computer directly and thoroughly to a task that the user wishes to perform. ... This article or section is in need of attention from an expert on the subject. ... For other uses, see Supercomputer (disambiguation). ... Distributed computing is a method of computer processing in which different parts of a program are run simultaneously on two or more computers that are communicating with each other over a network. ...

Numerical analysis is an important underpinning for techniques used in computational science. Numerical analysis is the study of approximate methods for the problems of continuous mathematics (as distinguished from discrete mathematics). ...

Applications of computational science

Problem domains for computational science/scientific computing include:

Numerical simulations

Numerical simulations have different objectives depending on the nature of the task being simulated:

• Reconstruct and understand known events (e.g., earthquake, tsunamis and other natural disasters).
• Predict future or unobserved situations (e.g., weather, sub-atomic particle behaviour).

Model fitting and data analysis

• Appropriately tune models or solve equations to reflect observations, subject to model constraints (e.g. oil exploration geophysics, computational linguistics)

• Use graph theory to model networks, especially those connecting individuals, organizations, and websites.

A drawing of a graph. ...

Optimization

• Optimize known scenarios (e.g., technical and manufacturing processes, front end engineering).

Methods and algorithms

Algorithms and mathematical methods used in computational science are varied. Commonly applied methods include:

• Numerical analysis
• Application of Taylor series as convergent and asymptotic series
• Computing derivatives by Automatic differentiation (AD)
• Computing derivatives by finite differences
• Graph theoretic suites
• High order difference approximations via Taylor series and Richardson extrapolation
• Methods for integration on a uniform mesh: rectangle rule, trapezoid rule, midpoint rule, Simpson's rule
• Runge Kutta method for solving ordinary differential equations
• Monte Carlo methods
• Numerical linear algebra
• Computing the LU factors by Gaussian elimination
• Choleski factorizations
• Discrete Fourier transform and applications.
• Newton's method
• Time stepping methods for dynamical systems

Programming languages commonly used for the more mathematical aspects of scientific computing applications include Mathematica, MATLAB, SciLab, GNU Octave, COMSOL Multiphysics, and PDL. The more computationally-intensive aspects of scientific computing will often utilize some variation of C or Fortran. Numerical analysis is the study of approximate methods for the problems of continuous mathematics (as distinguished from discrete mathematics). ... Series expansion redirects here. ... For the formal concept of computation, see computation. ... In mathematics and computer algebra, automatic differentiation, or AD, sometimes alternatively called algorithmic differentiation, is a method to numerically evaluate the derivative of a function specified by a computer program. ... For the formal concept of computation, see computation. ... There are two subfields of mathematics that concern themselves with finite differences. ... Series expansion redirects here. ... In numerical analysis, Richardson extrapolation is a sequence acceleration method, used to improve the rate of convergence of a sequence. ... In integral calculus, the rectangle method (also called the Mid-Ordinate Rule) uses an approximation to a definite integral, made by finding the area of a series of rectangles. ... The function f(x) (in blue) is approximated by a linear function (in red). ... The function f(x) (in blue) is approximated by a quadratic function P(x) (in red). ... In numerical analysis, the Runge-Kutta methods are an important family of implicit and explicit iterative methods for the approximation of solutions of ordinary differential equations. ... The Monte Carlo method can be illustrated as a game of battleship. ... Numerical linear algebra is often at the heart of many engineering and computational science problems, such as image and signal processing, computational finance, materials science simulations, structural biology, datamining, and bioinformatics just to name a few. ... For the formal concept of computation, see computation. ... In linear algebra, the LU decomposition is a matrix decomposition which writes a matrix as the product of a lower and upper triangular matrix. ... In linear algebra, Gaussian elimination is an algorithm that can be used to determine the solutions of a system of linear equations, to find the rank of a matrix, and to calculate the inverse of an invertible square matrix. ... In mathematics, the Cholesky decomposition, named after André-Louis Cholesky, is a matrix decomposition of a symmetric positive-definite matrix into a lower triangular matrix and the transpose of the lower triangular matrix. ... In mathematics, the discrete Fourier transform (DFT), occasionally called the finite Fourier transform, is a transform for Fourier analysis of finite-domain discrete-time signals. ... In numerical analysis, Newtons method (also known as the Newton–Raphson method or the Newton–Fourier method) is an efficient algorithm for finding approximations to the zeros (or roots) of a real-valued function. ... For other uses, see Mathematica (disambiguation). ... Not to be confused with Matlab Upazila in Chandpur District, Bangladesh. ... Scilab is a numerical computational package developed by INRIA and ENPC in France. ... Octave is a free computer program for performing numerical computations which is mostly compatible with MATLAB. It is part of the GNU project. ... COMSOL Multiphysics (formerly FEMLAB) is a finite element analysis and solver software package for various physics and engineering applications, especially coupled phenomena, or multiphysics. ... PDL (short for Perl Data Language) is a set of Array programming extensions to the Perl programming language. ... In computing, C is a general-purpose, block structured, procedural, imperative computer programming language developed in 1972 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system. ... Fortran (previously FORTRAN[1]) is a general-purpose[2], procedural,[3] imperative programming language that is especially suited to numeric computation and scientific computing. ...

Computational science application programs often model real-world changing conditions, such as weather, air flow around a plane, automobile body distortions in a crash, the motion of stars in a galaxy, an explosive device, etc. Such programs might create a 'logical mesh' in computer memory where each item corresponds to an area in space and contains information about that space relevant to the model. For example in weather models, each item might be a square kilometer; with land elevation, current wind direction, humidity, temperature, pressure, etc. The program would calculate the likely next state based on the current state, in simulated time steps, solving equations that describe how the system operates; and then repeat the process to calculate the next state.

The term computational scientist is used to describe someone skilled in scientific computing. This person is usually a scientist, an engineer or an applied mathematician who applies high-performance computers in different ways to advance the state-of-the-art in their respective applied disciplines in physics, chemistry or engineering. Scientific computing has increasingly also impacted on other areas including economics, biology and medicine. The term computational scientist is used to describe someone skilled in scientific computing. ...

Computational science is now commonly considered a third mode of science, complementing and adding to experimentation/observation and theory. [1] A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... From Latin ex- + -periri (akin to periculum attempt). ... For other uses, see Observation (disambiguation). ... The word theory has a number of distinct meanings in different fields of knowledge, depending on their methodologies and the context of discussion. ...

Education

Scientific computation is most often studied through an applied mathematics or computer science program, or within a standard mathematics, sciences, or engineering program. At some institutions a specialization in scientific computation can be earned as a "minor" within another program (which may be at varying levels). However, there are increasingly many bachelor's and master's programs in computational science. Some schools also offer the Ph.D. in computational science, computational engineering, computational science and engineering, or scientific computation. Check External Links below for universities that offer computational science programs. Applied mathematics is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains. ... Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ... A bachelors degree is usually an undergraduate academic degree awarded for a course or major that generally lasts for three, four, or in some cases and countries, five or six years. ... It has been suggested that this article or section be merged with Computational Science. ... // Computational science (or scientific computing) is the field of study concerned with constructing mathematical models and numerical solution techniques and using computers to analyse and solve scientific, social scientific and engineering problems. ...

There are also programs in areas such as computational physics, computational chemistry, etc. Computational physics is the study and implementation of numerical algorithms in order to solve problems in physics for which a quantitative theory already exists. ... Computational chemistry is a branch of chemistry that uses the results of theoretical chemistry incorporated into efficient computer programs to calculate the structures and properties of molecules and solids, applying these programs to complement the information obtained by actual chemical experiments, predict hitherto unobserved chemical phenomena, and solve related problems. ...

Related fields

Map of the human X chromosome (from the NCBI website). ... Cheminformatics (also known as chemoinformatics and chemical informatics) is the use of computer and informational techniques, applied to a range of problems in the field of chemistry. ... Chemometrics is the application of mathematical or statistical methods to chemical data. ... Computational biology is an interdisciplinary field that applies the techniques of computer science and applied mathematics to problems inspired by biology. ... Computational chemistry is a branch of chemistry that uses the results of theoretical chemistry incorporated into efficient computer programs to calculate the structures and properties of molecules and solids, applying these programs to complement the information obtained by actual chemical experiments, predict hitherto unobserved chemical phenomena, and solve related problems. ... Computational economics is a form of economics which relies on mathematical methods, including mathematical economics and econometrics. ... Computational electromagnetics, computational electrodynamics or electromagnetic modeling refers to the process of modeling the interaction of electromagnetic fields with physical objects and the environment. ... It has been suggested that this article or section be merged with Computational Science. ... Computational finance (also known as financial engineering) is a cross-disciplinary field which relies on mathematical finance, numerical methods and computer simulations to make trading, hedging and investment decisions, as well as facilitating the risk management of those decisions. ... A computer simulation of high velocity air flow around the Space Shuttle during re-entry. ... Computational mathematics involves mathematical research in areas of science where computing plays a central and essential role, emphasizing algorithms, numerical methods, and symbolic methods. ... Computational mechanics is the subject/profession concerned with the use of computational methods and devices to study phenomena governed by the principles of mechanics. ... Computational particle physics refers to the methods and computing tools developed in and used by particle physics research. ... Computational physics is the study and implementation of numerical algorithms in order to solve problems in physics for which a quantitative theory already exists. ... Computational statistics, or statistical computing, is the interface between statistics, computer science and numerical analysis. ... A computer algebra system (CAS) is a software program that facilitates symbolic mathematics. ... Computation of corporate finance problems, standard portfolio problems, option pricing and applications, and duration and immunization. ... GIS redirects here. ... The field of high performance computing (HPC) comprises computing applications on (parallel) supercomputers and computer clusters. ... As a broad subfield of artificial intelligence, machine learning is concerned with the design and development of algorithms and techniques that allow computers to learn. At a general level, there are two types of learning: inductive, and deductive. ... Network analysis is the analysis of networks through network theory (or more generally graph theory). ... An example of 500 mb geopotential height prediction from a numerical weather prediction model Numerical weather prediction is the science of predicting the weather using mathematical models of the atmosphere. ... Pattern recognition is a field within the area of machine learning. ...

See also

• List of numerical analysis software
• Comparison of computer algebra systems
• List of statistical packages
• List of molecular modeling software

Listed here are a number of computer programs used for performing numerical calculations: Baudline is a time-frequency browser for numerical signals analysis and scientific visualization. ... The following tables provide a comparison of computer algebra systems (CAS). ... This is an incomplete list of software that is designed for the explicit purpose of performing statistical analyses. ... Min - Optimization, MD - Molecular Dynamics, MC - Monte Carlo, QM - Quantum mechanics. ...

External links

• Links to Downloadable Computational Tools
• SIAM Journal on Scientific Computing
• Computing in Science & Engineering magazine
• Scientific Computing magazine
• Educational Materials for Undergraduate Computational Studies
• Brockport State College Computational Science B.S. program, with reports
• University of Amsterdam International Master Computational Science, MSc program