1.
Scientific visualization
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Scientific visualization is an interdisciplinary branch of science. It is also considered a subset of computer graphics, a branch of computer science, the purpose of scientific visualization is to graphically illustrate scientific data to enable scientists to understand, illustrate, and glean insight from their data. One of the earliest examples of scientific visualisation was Maxwells thermodynamic surface. This prefigured modern scientific techniques that use computer graphics. Scientific visualization using computer graphics gained in popularity as graphics matured, primary applications were scalar fields and vector fields from computer simulations and also measured data. The primary methods for visualizing two-dimensional scalar fields are color mapping and drawing contour lines, 2D vector fields are visualized using glyphs and streamlines or line integral convolution methods. For 3D scalar fields the primary methods are volume rendering and isosurfaces, methods for visualizing vector fields include glyphs such as arrows, streamlines and streaklines, particle tracing, line integral convolution and topological methods. Later, visualization techniques such as hyperstreamlines were developed to visualize 2D, computer animation is the art, technique, and science of creating moving images via the use of computers. It is becoming common to be created by means of 3D computer graphics, though 2D computer graphics are still widely used for stylistic, low bandwidth. Sometimes the target of the animation is the computer itself, but sometimes the target is another medium and it is also referred to as CGI, especially when used in films. Computer simulation is a program, or network of computers. The simultaneous visualization and simulation of a system is called visulation, computer simulations vary from computer programs that run a few minutes, to network-based groups of computers running for hours, to ongoing simulations that run for months. Information visualization focused on the creation of approaches for conveying information in intuitive ways. The key difference between scientific visualization and information visualization is that information visualization is often applied to data that is not generated by scientific inquiry, some examples are graphical representations of data for business, government, news and social media. Interface technology and perception shows how new interfaces and an understanding of underlying perceptual issues create new opportunities for the scientific visualization community. Rendering is the process of generating an image from a model, the model is a description of three-dimensional objects in a strictly defined language or data structure. It would contain geometry, viewpoint, texture, lighting, the image is a digital image or raster graphics image. The term may be by analogy with a rendering of a scene
2.
Computational fluid dynamics
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Computational fluid dynamics is a branch of fluid mechanics that uses numerical analysis and data structures to solve and analyze problems that involve fluid flows. Computers are used to perform the required to simulate the interaction of liquids. With high-speed supercomputers, better solutions can be achieved, ongoing research yields software that improves the accuracy and speed of complex simulation scenarios such as transonic or turbulent flows. Initial experimental validation of such software is performed using a tunnel with the final validation coming in full-scale testing. The fundamental basis of almost all CFD problems is the Navier–Stokes equations and these equations can be simplified by removing terms describing viscous actions to yield the Euler equations. Further simplification, by removing terms describing vorticity yields the full potential equations, finally, for small perturbations in subsonic and supersonic flows these equations can be linearized to yield the linearized potential equations. Historically, methods were first developed to solve the potential equations. Two-dimensional methods, using conformal transformations of the flow about a cylinder to the flow about an airfoil were developed in the 1930s. One of the earliest type of calculations resembling modern CFD are those by Lewis Fry Richardson, in the sense that these calculations used finite differences and divided the physical space in cells. Although they failed dramatically, these calculations, together with Richardsons book Weather prediction by numerical process, set the basis for modern CFD, in fact, early CFD calculations during the 1940s using ENIAC used methods close to those in Richardsons 1922 book. The computer power available paced development of three-dimensional methods, probably the first work using computers to model fluid flow, as governed by the Navier-Stokes equations, was performed at Los Alamos National Lab, in the T3 group. This group was led by Francis H. Harlow, who is considered as one of the pioneers of CFD. Fromms vorticity-stream-function method for 2D, transient, incompressible flow was the first treatment of strongly contorting incompressible flows in the world, the first paper with three-dimensional model was published by John Hess and A. M. O. Smith of Douglas Aircraft in 1967 and this method discretized the surface of the geometry with panels, giving rise to this class of programs being called Panel Methods. Their method itself was simplified, in that it did not include lifting flows and hence was mainly applied to ship hulls, the first lifting Panel Code was described in a paper written by Paul Rubbert and Gary Saaris of Boeing Aircraft in 1968. In time, more advanced three-dimensional Panel Codes were developed at Boeing, Lockheed, Douglas, McDonnell Aircraft, NASA, some were higher order codes, using higher order distributions of surface singularities, while others used single singularities on each surface panel. The advantage of the lower order codes was that they ran much faster on the computers of the time, today, VSAERO has grown to be a multi-order code and is the most widely used program of this class. It has been used in the development of submarines, surface ships, automobiles, helicopters, aircraft
3.
Stanislaw Ulam
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Stanisław Marcin Ulam was a Polish-American mathematician. In pure and applied mathematics, he proved some theorems and proposed several conjectures, born into a wealthy Polish Jewish family, Ulam studied mathematics at the Lwów Polytechnic Institute, where he earned his PhD in 1933 under the supervision of Kazimierz Kuratowski. In 1935, John von Neumann, whom Ulam had met in Warsaw, invited him to come to the Institute for Advanced Study in Princeton, New Jersey, for a few months. From 1936 to 1939, he spent summers in Poland and academic years at Harvard University in Cambridge, Massachusetts, on 20 August 1939, he sailed for America for the last time with his 17-year-old brother Adam Ulam. He became an assistant professor at the University of Wisconsin–Madison in 1940, in October 1943, he received an invitation from Hans Bethe to join the Manhattan Project at the secret Los Alamos Laboratory in New Mexico. There, he worked on the calculations to predict the behavior of the explosive lenses that were needed by an implosion-type weapon. He was assigned to Edward Tellers group, where he worked on Tellers Super bomb for Teller, after the war he left to become an associate professor at the University of Southern California, but returned to Los Alamos in 1946 to work on thermonuclear weapons. With the aid of a cadre of female computers, including his wife Françoise Aron Ulam, in January 1951, Ulam and Teller came up with the Teller–Ulam design, which is the basis for all thermonuclear weapons. With Fermi and John Pasta, Ulam studied the Fermi–Pasta–Ulam problem, Ulam was born in Lemberg, Galicia, on 13 April 1909. At this time, Galicia was in the Kingdom of Galicia and Lodomeria of the Austro-Hungarian Empire, in 1918, it became part of the newly restored Poland, the Second Polish Republic, and the city took its Polish name again, Lwów. The Ulams were a wealthy Polish Jewish family of bankers, industrialists, Ulams immediate family was well-to-do but hardly rich. His father, Józef Ulam, was born in Lwów and was a lawyer and his uncle, Michał Ulam, was an architect, building contractor, and lumber industrialist. From 1916 until 1918, Józefs family lived temporarily in Vienna, after they returned, Lwów became the epicenter of the Polish–Ukrainian War, during which the city experienced a Ukrainian siege. In 1919, Ulam entered Lwów Gymnasium Nr, VII, from which he graduated in 1927. He then studied mathematics at the Lwów Polytechnic Institute, under the supervision of Kazimierz Kuratowski, he received his Master of Arts degree in 1932, and became a Doctor of Science in 1933. At the age of 20, in 1929, he published his first paper Concerning Function of Sets in the journal Fundamenta Mathematicae. From 1931 until 1935, he traveled to and studied in Wilno, Vienna, Zurich, Paris, and Cambridge, England, along with Stanisław Mazur, Mark Kac, Włodzimierz Stożek, Kuratowski, and others, Ulam was a member of the Lwów School of Mathematics. Its founders were Hugo Steinhaus and Stefan Banach, who were professors at the University of Lwów, mathematicians of this school met for long hours at the Scottish Café, where the problems they discussed were collected in the Scottish Book, a thick notebook provided by Banachs wife
4.
Physics
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Physics is the natural science that involves the study of matter and its motion and behavior through space and time, along with related concepts such as energy and force. One of the most fundamental disciplines, the main goal of physics is to understand how the universe behaves. Physics is one of the oldest academic disciplines, perhaps the oldest through its inclusion of astronomy, Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the mechanisms of other sciences while opening new avenues of research in areas such as mathematics. Physics also makes significant contributions through advances in new technologies that arise from theoretical breakthroughs, the United Nations named 2005 the World Year of Physics. Astronomy is the oldest of the natural sciences, the stars and planets were often a target of worship, believed to represent their gods. While the explanations for these phenomena were often unscientific and lacking in evidence, according to Asger Aaboe, the origins of Western astronomy can be found in Mesopotamia, and all Western efforts in the exact sciences are descended from late Babylonian astronomy. The most notable innovations were in the field of optics and vision, which came from the works of many scientists like Ibn Sahl, Al-Kindi, Ibn al-Haytham, Al-Farisi and Avicenna. The most notable work was The Book of Optics, written by Ibn Al-Haitham, in which he was not only the first to disprove the ancient Greek idea about vision, but also came up with a new theory. In the book, he was also the first to study the phenomenon of the pinhole camera, many later European scholars and fellow polymaths, from Robert Grosseteste and Leonardo da Vinci to René Descartes, Johannes Kepler and Isaac Newton, were in his debt. Indeed, the influence of Ibn al-Haythams Optics ranks alongside that of Newtons work of the same title, the translation of The Book of Optics had a huge impact on Europe. From it, later European scholars were able to build the devices as what Ibn al-Haytham did. From this, such important things as eyeglasses, magnifying glasses, telescopes, Physics became a separate science when early modern Europeans used experimental and quantitative methods to discover what are now considered to be the laws of physics. Newton also developed calculus, the study of change, which provided new mathematical methods for solving physical problems. The discovery of new laws in thermodynamics, chemistry, and electromagnetics resulted from greater research efforts during the Industrial Revolution as energy needs increased, however, inaccuracies in classical mechanics for very small objects and very high velocities led to the development of modern physics in the 20th century. Modern physics began in the early 20th century with the work of Max Planck in quantum theory, both of these theories came about due to inaccuracies in classical mechanics in certain situations. Quantum mechanics would come to be pioneered by Werner Heisenberg, Erwin Schrödinger, from this early work, and work in related fields, the Standard Model of particle physics was derived. Areas of mathematics in general are important to this field, such as the study of probabilities, in many ways, physics stems from ancient Greek philosophy
5.
Numerical analysis
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Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis. Being able to compute the sides of a triangle is important, for instance, in astronomy, carpentry. Numerical analysis continues this tradition of practical mathematical calculations. Much like the Babylonian approximation of the root of 2, modern numerical analysis does not seek exact answers. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors, before the advent of modern computers numerical methods often depended on hand interpolation in large printed tables. Since the mid 20th century, computers calculate the required functions instead and these same interpolation formulas nevertheless continue to be used as part of the software algorithms for solving differential equations. Computing the trajectory of a spacecraft requires the accurate numerical solution of a system of differential equations. Car companies can improve the safety of their vehicles by using computer simulations of car crashes. Such simulations essentially consist of solving differential equations numerically. Hedge funds use tools from all fields of analysis to attempt to calculate the value of stocks. Airlines use sophisticated optimization algorithms to decide ticket prices, airplane and crew assignments, historically, such algorithms were developed within the overlapping field of operations research. Insurance companies use programs for actuarial analysis. The rest of this section outlines several important themes of numerical analysis, the field of numerical analysis predates the invention of modern computers by many centuries. Linear interpolation was already in use more than 2000 years ago, to facilitate computations by hand, large books were produced with formulas and tables of data such as interpolation points and function coefficients. The function values are no very useful when a computer is available. The mechanical calculator was developed as a tool for hand computation. These calculators evolved into electronic computers in the 1940s, and it was found that these computers were also useful for administrative purposes. But the invention of the computer also influenced the field of analysis, since now longer
6.
John von Neumann
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John von Neumann was a Hungarian-American mathematician, physicist, inventor, computer scientist, and polymath. He made major contributions to a number of fields, including mathematics, physics, economics, computing, and statistics. He published over 150 papers in his life, about 60 in pure mathematics,20 in physics, and 60 in applied mathematics and his last work, an unfinished manuscript written while in the hospital, was later published in book form as The Computer and the Brain. His analysis of the structure of self-replication preceded the discovery of the structure of DNA, also, my work on various forms of operator theory, Berlin 1930 and Princeton 1935–1939, on the ergodic theorem, Princeton, 1931–1932. During World War II he worked on the Manhattan Project, developing the mathematical models behind the lenses used in the implosion-type nuclear weapon. After the war, he served on the General Advisory Committee of the United States Atomic Energy Commission, along with theoretical physicist Edward Teller, mathematician Stanislaw Ulam, and others, he worked out key steps in the nuclear physics involved in thermonuclear reactions and the hydrogen bomb. Von Neumann was born Neumann János Lajos to a wealthy, acculturated, Von Neumanns place of birth was Budapest in the Kingdom of Hungary which was then part of the Austro-Hungarian Empire. He was the eldest of three children and he had two younger brothers, Michael, born in 1907, and Nicholas, who was born in 1911. His father, Neumann Miksa was a banker, who held a doctorate in law and he had moved to Budapest from Pécs at the end of the 1880s. Miksas father and grandfather were both born in Ond, Zemplén County, northern Hungary, johns mother was Kann Margit, her parents were Jakab Kann and Katalin Meisels. Three generations of the Kann family lived in apartments above the Kann-Heller offices in Budapest. In 1913, his father was elevated to the nobility for his service to the Austro-Hungarian Empire by Emperor Franz Joseph, the Neumann family thus acquired the hereditary appellation Margittai, meaning of Marghita. The family had no connection with the town, the appellation was chosen in reference to Margaret, Neumann János became Margittai Neumann János, which he later changed to the German Johann von Neumann. Von Neumann was a child prodigy, as a 6 year old, he could multiply and divide two 8-digit numbers in his head, and could converse in Ancient Greek. When he once caught his mother staring aimlessly, the 6 year old von Neumann asked her, formal schooling did not start in Hungary until the age of ten. Instead, governesses taught von Neumann, his brothers and his cousins, Max believed that knowledge of languages other than Hungarian was essential, so the children were tutored in English, French, German and Italian. A copy was contained in a private library Max purchased, One of the rooms in the apartment was converted into a library and reading room, with bookshelves from ceiling to floor. Von Neumann entered the Lutheran Fasori Evangelikus Gimnázium in 1911 and this was one of the best schools in Budapest, part of a brilliant education system designed for the elite
7.
Chaos theory
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Chaos theory is a branch of mathematics focused on the behavior of dynamical systems that are highly sensitive to initial conditions. This happens even though these systems are deterministic, meaning that their behavior is fully determined by their initial conditions. In other words, the nature of these systems does not make them predictable. This behavior is known as chaos, or simply chaos. The theory was summarized by Edward Lorenz as, Chaos, When the present determines the future, Chaotic behavior exists in many natural systems, such as weather and climate. It also occurs spontaneously in some systems with components, such as road traffic. This behavior can be studied through analysis of a mathematical model, or through analytical techniques such as recurrence plots. Chaos theory has applications in several disciplines, including meteorology, sociology, physics, environmental science, computer science, engineering, economics, biology, ecology, the theory formed the basis for such fields of study as complex dynamical systems, edge of chaos theory, self-assembly process. Chaos theory concerns deterministic systems whose behavior can in principle be predicted, Chaotic systems are predictable for a while and then appear to become random. Some examples of Lyapunov times are, chaotic electrical circuits, about 1 millisecond, weather systems, a few days, in chaotic systems, the uncertainty in a forecast increases exponentially with elapsed time. Hence, mathematically, doubling the forecast time more than squares the proportional uncertainty in the forecast and this means, in practice, a meaningful prediction cannot be made over an interval of more than two or three times the Lyapunov time. When meaningful predictions cannot be made, the system appears random, in common usage, chaos means a state of disorder. However, in theory, the term is defined more precisely. Although no universally accepted mathematical definition of chaos exists, a commonly used definition originally formulated by Robert L, in these cases, while it is often the most practically significant property, sensitivity to initial conditions need not be stated in the definition. If attention is restricted to intervals, the second property implies the other two, an alternative, and in general weaker, definition of chaos uses only the first two properties in the above list. Sensitivity to initial conditions means that each point in a system is arbitrarily closely approximated by other points with significantly different future paths. Thus, a small change, or perturbation, of the current trajectory may lead to significantly different future behavior. C. Entitled Predictability, Does the Flap of a Butterflys Wings in Brazil set off a Tornado in Texas, the flapping wing represents a small change in the initial condition of the system, which causes a chain of events leading to large-scale phenomena
8.
Stark effect
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The Stark effect is the shifting and splitting of spectral lines of atoms and molecules due to presence of an external electric field. The amount of splitting or shifting is called the Stark splitting or Stark shift, in general, one distinguishes first- and second-order Stark effects. The first-order effect is linear in the electric field, while the second-order effect is quadratic in the field. The Stark effect is responsible for the broadening of spectral lines by charged particles. When the split/shifted lines appear in absorption, the effect is called the inverse Stark effect, the Stark effect is the electric analogue of the Zeeman effect where a spectral line is split into several components due to the presence of a magnetic field. The Stark effect can be explained with fully quantum-mechanical approaches, the effect is named after Johannes Stark, who discovered it in 1913. It was independently discovered in the year by the Italian physicist Antonino Lo Surdo. The discovery of this effect contributed importantly to the development of quantum theory, by using experimental indices of refraction he gave an estimate of the Stark splittings. This estimate was a few orders of magnitude too low, not deterred by this prediction, Stark undertook measurements on excited states of the hydrogen atom and succeeded in observing splittings. By the use of the Bohr–Sommerfeld quantum theory, Paul Epstein and Karl Schwarzschild were independently able to derive equations for the linear, four years later, Hendrik Kramers derived formulas for intensities of spectral transitions. Kramers also included the effect of structure, which includes corrections for relativistic kinetic energy. The first quantum mechanical treatment was by Wolfgang Pauli, erwin Schrödinger discussed at length the Stark effect in his third paper on quantum theory, once in the manner of the 1916 work of Epstein and once by his perturbation approach. Finally, Epstein reconsidered the linear and quadratic Stark effect from the point of view of the new quantum theory and he derived equations for the line intensities which were a decided improvement over Kramers results obtained by the old quantum theory. While first-order perturbation effects for the Stark effect in hydrogen are in agreement for the Bohr–Sommerfeld model, measurements of the Stark effect under high field strengths confirmed the correctness of the quantum theory over the Bohr model. An electric field pointing from left to right, for example, tends to pull nuclei to the right and electrons to the left. Other things equal, the effect of the field is greater for outer electron shells, because the electron is more distant from the nucleus, so it travels farther left. The Stark effect can lead to splitting of energy levels. For example, in the Bohr model, an electron has the same whether it is in the 2s state or any of the 2p states
9.
N-body problem
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In physics, the n-body problem is the problem of predicting the individual motions of a group of celestial objects interacting with each other gravitationally. Solving this problem has been motivated by the desire to understand the motions of the Sun, Moon, planets, in the 20th century, understanding the dynamics of globular cluster star systems became an important n-body problem. The n-body problem in general relativity is more difficult to solve. Having done so, he and others soon discovered over the course of a few years, Newton realized it was because gravitational interactive forces amongst all the planets was affecting all their orbits. Thus came the awareness and rise of the problem in the early 17th century. Ironically, this conformity led to the wrong approach, after Newtons time the n-body problem historically was not stated correctly because it did not include a reference to those gravitational interactive forces. Newton does not say it directly but implies in his Principia the n-body problem is unsolvable because of gravitational interactive forces. Newton said in his Principia, paragraph 21, And hence it is that the force is found in both bodies. The Sun attracts Jupiter and the planets, Jupiter attracts its satellites. Two bodies can be drawn to other by the contraction of rope between them. Newton concluded via his third law of motion according to this Law all bodies must attract each other. This last statement, which implies the existence of gravitational forces, is key. The problem of finding the solution of the n-body problem was considered very important. Indeed, in the late 19th century King Oscar II of Sweden, advised by Gösta Mittag-Leffler, in case the problem could not be solved, any other important contribution to classical mechanics would then be considered to be prizeworthy. The prize was awarded to Poincaré, even though he did not solve the original problem, the version finally printed contained many important ideas which led to the development of chaos theory. The problem as stated originally was finally solved by Karl Fritiof Sundman for n =3. The n-body problem considers n point masses mi, i =1,2, …, n in a reference frame in three dimensional space ℝ3 moving under the influence of mutual gravitational attraction. Each mass mi has a position vector qi, Newtons second law says that mass times acceleration mi d2qi/dt2 is equal to the sum of the forces on the mass
10.
Finite element method
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The finite element method is a numerical method for solving problems of engineering and mathematical physics. It is also referred to as finite element analysis, typical problem areas of interest include structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. The analytical solution of these problems generally require the solution to boundary value problems for partial differential equations, the finite element method formulation of the problem results in a system of algebraic equations. The method yields approximate values of the unknowns at discrete number of points over the domain, to solve the problem, it subdivides a large problem into smaller, simpler parts that are called finite elements. The simple equations that model these finite elements are assembled into a larger system of equations that models the entire problem. FEM then uses variational methods from the calculus of variations to approximate a solution by minimizing an associated error function, the global system of equations has known solution techniques, and can be calculated from the initial values of the original problem to obtain a numerical answer. To explain the approximation in this process, FEM is commonly introduced as a case of Galerkin method. The process, in language, is to construct an integral of the inner product of the residual. In simple terms, it is a procedure that minimizes the error of approximation by fitting trial functions into the PDE, the residual is the error caused by the trial functions, and the weight functions are polynomial approximation functions that project the residual. These equation sets are the element equations and they are linear if the underlying PDE is linear, and vice versa. In step above, a system of equations is generated from the element equations through a transformation of coordinates from the subdomains local nodes to the domains global nodes. This spatial transformation includes appropriate orientation adjustments as applied in relation to the coordinate system. The process is carried out by FEM software using coordinate data generated from the subdomains. FEM is best understood from its application, known as finite element analysis. FEA as applied in engineering is a tool for performing engineering analysis. It includes the use of mesh generation techniques for dividing a complex problem into small elements, FEA is a good choice for analyzing problems over complicated domains, when the domain changes, when the desired precision varies over the entire domain, or when the solution lacks smoothness. For instance, in a crash simulation it is possible to increase prediction accuracy in important areas like the front of the car. Another example would be in weather prediction, where it is more important to have accurate predictions over developing highly nonlinear phenomena rather than relatively calm areas
11.
Algorithm
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In mathematics and computer science, an algorithm is a self-contained sequence of actions to be performed. Algorithms can perform calculation, data processing and automated reasoning tasks, an algorithm is an effective method that can be expressed within a finite amount of space and time and in a well-defined formal language for calculating a function. The transition from one state to the next is not necessarily deterministic, some algorithms, known as randomized algorithms, giving a formal definition of algorithms, corresponding to the intuitive notion, remains a challenging problem. In English, it was first used in about 1230 and then by Chaucer in 1391, English adopted the French term, but it wasnt until the late 19th century that algorithm took on the meaning that it has in modern English. Another early use of the word is from 1240, in a manual titled Carmen de Algorismo composed by Alexandre de Villedieu and it begins thus, Haec algorismus ars praesens dicitur, in qua / Talibus Indorum fruimur bis quinque figuris. Which translates as, Algorism is the art by which at present we use those Indian figures, the poem is a few hundred lines long and summarizes the art of calculating with the new style of Indian dice, or Talibus Indorum, or Hindu numerals. An informal definition could be a set of rules that precisely defines a sequence of operations, which would include all computer programs, including programs that do not perform numeric calculations. Generally, a program is only an algorithm if it stops eventually, but humans can do something equally useful, in the case of certain enumerably infinite sets, They can give explicit instructions for determining the nth member of the set, for arbitrary finite n. An enumerably infinite set is one whose elements can be put into one-to-one correspondence with the integers, the concept of algorithm is also used to define the notion of decidability. That notion is central for explaining how formal systems come into being starting from a set of axioms. In logic, the time that an algorithm requires to complete cannot be measured, from such uncertainties, that characterize ongoing work, stems the unavailability of a definition of algorithm that suits both concrete and abstract usage of the term. Algorithms are essential to the way computers process data, thus, an algorithm can be considered to be any sequence of operations that can be simulated by a Turing-complete system. Although this may seem extreme, the arguments, in its favor are hard to refute. Gurevich. Turings informal argument in favor of his thesis justifies a stronger thesis, according to Savage, an algorithm is a computational process defined by a Turing machine. Typically, when an algorithm is associated with processing information, data can be read from a source, written to an output device. Stored data are regarded as part of the state of the entity performing the algorithm. In practice, the state is stored in one or more data structures, for some such computational process, the algorithm must be rigorously defined, specified in the way it applies in all possible circumstances that could arise. That is, any conditional steps must be dealt with, case-by-case
12.
Computer simulation
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Computer simulations reproduce the behavior of a system using a model. Simulation of a system is represented as the running of the systems model and it can be used to explore and gain new insights into new technology and to estimate the performance of systems too complex for analytical solutions. The scale of events being simulated by computer simulations has far exceeded anything possible using traditional paper-and-pencil mathematical modeling, other examples include a 1-billion-atom model of material deformation, a 2. Because of the computational cost of simulation, computer experiments are used to perform such as uncertainty quantification. A computer model is the algorithms and equations used to capture the behavior of the system being modeled, by contrast, computer simulation is the actual running of the program that contains these equations or algorithms. Simulation, therefore, is the process of running a model, thus one would not build a simulation, instead, one would build a model, and then either run the model or equivalently run a simulation. It was a simulation of 12 hard spheres using a Monte Carlo algorithm, Computer simulation is often used as an adjunct to, or substitute for, modeling systems for which simple closed form analytic solutions are not possible. The external data requirements of simulations and models vary widely, for some, the input might be just a few numbers, while others might require terabytes of information. Because of this variety, and because diverse simulation systems have common elements. Systems that accept data from external sources must be careful in knowing what they are receiving. While it is easy for computers to read in values from text or binary files, often they are expressed as error bars, a minimum and maximum deviation from the value range within which the true value lie. Even small errors in the data can accumulate into substantial error later in the simulation. While all computer analysis is subject to the GIGO restriction, this is true of digital simulation. Indeed, observation of this inherent, cumulative error in digital systems was the main catalyst for the development of chaos theory, another way of categorizing models is to look at the underlying data structures. For time-stepped simulations, there are two classes, Simulations which store their data in regular grids and require only next-neighbor access are called stencil codes. Many CFD applications belong to this category, if the underlying graph is not a regular grid, the model may belong to the meshfree method class. Equations define the relationships between elements of the system and attempt to find a state in which the system is in equilibrium. Such models are used in simulating physical systems, as a simpler modeling case before dynamic simulation is attempted
