Computer simulation

Computer simulation is the reproduction of the behavior of a system using a computer to simulate the outcomes of a mathematical model associated with said system. Since they allow to check the reliability of chosen mathematical models, computer simulations have become a useful tool for the mathematical modeling of many natural systems in physics, climatology, chemistry and manufacturing, human systems in economics, social science, health care and engineering. Simulation of a system is represented as the running of the system's model, 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. Computer simulations are realized by running computer programs that can be either small, running instantly on small devices, or large-scale programs that run for hours or days on network-based groups of computers; the scale of events being simulated by computer simulations has far exceeded anything possible using traditional paper-and-pencil mathematical modeling.

Over 10 years ago, a desert-battle simulation of one force invading another involved the modeling of 66,239 tanks and other vehicles on simulated terrain around Kuwait, using multiple supercomputers in the DoD High Performance Computer Modernization Program. Other examples include a 1-billion-atom model of material deformation; because of the computational cost of simulation, computer experiments are used to perform inference 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". Computer simulation developed hand-in-hand with the rapid growth of the computer, following its first large-scale deployment during the Manhattan Project in World War II to model the process of nuclear detonation, it was a simulation of 12 hard spheres using a Monte Carlo algorithm.

Computer simulation is used as an adjunct to, or substitute for, modeling systems for which simple closed form analytic solutions are not possible. There are many types of computer simulations; 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. Input sources vary widely: Sensors and other physical devices connected to the model. Lastly, the time at which data is available varies: "invariant" data is built into the model code, either because the value is invariant or because the designers consider the value to be invariant for all cases of interest; because of this variety, because diverse simulation systems have many common elements, there are a large number of specialized simulation languages. The best-known may be Simula. There are now many others. 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, what is much harder is knowing what the accuracy of the values are.

They are expressed as "error bars", a minimum and maximum deviation from the value range within which the true value lie. Because digital computer mathematics is not perfect and truncation errors multiply this error, so it is useful to perform an "error analysis" to confirm that values output by the simulation will still be usefully accurate. Small errors in the original data can accumulate into substantial error 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. Computer models can be classified according to several independent pairs of attributes, including: Stochastic or deterministic – see external links below for examples of stochastic vs. deterministic simulations Steady-state or dynamic Continuous or discrete Dynamic system simulation, e.g. electric systems, hydraulic systems or multi-body mechanical s

Numerical weather prediction

Numerical weather prediction uses mathematical models of the atmosphere and oceans to predict the weather based on current weather conditions. Though first attempted in the 1920s, it was not until the advent of computer simulation in the 1950s that numerical weather predictions produced realistic results. A number of global and regional forecast models are run in different countries worldwide, using current weather observations relayed from radiosondes, weather satellites and other observing systems as inputs. Mathematical models based on the same physical principles can be used to generate either short-term weather forecasts or longer-term climate predictions; the improvements made to regional models have allowed for significant improvements in tropical cyclone track and air quality forecasts. Manipulating the vast datasets and performing the complex calculations necessary to modern numerical weather prediction requires some of the most powerful supercomputers in the world. With the increasing power of supercomputers, the forecast skill of numerical weather models extends to only about six days.

Factors affecting the accuracy of numerical predictions include the density and quality of observations used as input to the forecasts, along with deficiencies in the numerical models themselves. Post-processing techniques such as model output statistics have been developed to improve the handling of errors in numerical predictions. A more fundamental problem lies in the chaotic nature of the partial differential equations that govern the atmosphere, it is impossible to solve these equations and small errors grow with time. Present understanding is that this chaotic behavior limits accurate forecasts to about 14 days with accurate input data and a flawless model. In addition, the partial differential equations used in the model need to be supplemented with parameterizations for solar radiation, moist processes, heat exchange, vegetation, surface water, the effects of terrain. In an effort to quantify the large amount of inherent uncertainty remaining in numerical predictions, ensemble forecasts have been used since the 1990s to help gauge the confidence in the forecast, to obtain useful results farther into the future than otherwise possible.

This approach analyzes multiple forecasts created with an individual forecast model or multiple models. The history of numerical weather prediction began in the 1920s through the efforts of Lewis Fry Richardson, who used procedures developed by Vilhelm Bjerknes to produce by hand a six-hour forecast for the state of the atmosphere over two points in central Europe, taking at least six weeks to do so, it was not until the advent of the computer and computer simulations that computation time was reduced to less than the forecast period itself. The ENIAC was used to create the first weather forecasts via computer in 1950, based on a simplified approximation to the atmospheric governing equations. In 1954, Carl-Gustav Rossby's group at the Swedish Meteorological and Hydrological Institute used the same model to produce the first operational forecast. Operational numerical weather prediction in the United States began in 1955 under the Joint Numerical Weather Prediction Unit, a joint project by the U.

S. Air Force and Weather Bureau. In 1956, Norman Phillips developed a mathematical model which could realistically depict monthly and seasonal patterns in the troposphere. Following Phillips' work, several groups began working to create general circulation models; the first general circulation climate model that combined both oceanic and atmospheric processes was developed in the late 1960s at the NOAA Geophysical Fluid Dynamics Laboratory. As computers have become more powerful, the size of the initial data sets has increased and newer atmospheric models have been developed to take advantage of the added available computing power; these newer models include more physical processes in the simplifications of the equations of motion in numerical simulations of the atmosphere. In 1966, West Germany and the United States began producing operational forecasts based on primitive-equation models, followed by the United Kingdom in 1972 and Australia in 1977; the development of limited area models facilitated advances in forecasting the tracks of tropical cyclones as well as air quality in the 1970s and 1980s.

By the early 1980s models began to include the interactions of soil and vegetation with the atmosphere, which led to more realistic forecasts. The output of forecast models based on atmospheric dynamics is unable to resolve some details of the weather near the Earth's surface; as such, a statistical relationship between the output of a numerical weather model and the ensuing conditions at the ground was developed in the 1970s and 1980s, known as model output statistics. Starting in the 1990s, model ensemble forecasts have been used to help define the forecast uncertainty and to extend the window in which numerical weather forecasting is viable farther into the future than otherwise possible; the atmosphere is a fluid. As such, the idea of numerical weather prediction is to sample the state of the fluid at a given time and use the equations of fluid dynamics and thermodynamics to estimate the state of the fluid at some time in the future; the process of entering observation data into the model to generate initial conditions is called initialization.

On land, terrain maps available at resolutions dow

Lorenz system

The Lorenz system is a system of ordinary differential equations first studied by Edward Lorenz. It is notable for having chaotic solutions for initial conditions. In particular, the Lorenz attractor is a set of chaotic solutions of the Lorenz system which, when plotted, resemble a butterfly or figure eight. In 1963, Edward Lorenz developed a simplified mathematical model for atmospheric convection; the model is a system of three ordinary differential equations now known as the Lorenz equations: d x d t = σ, d y d t = x − y, d z d t = x y − β z. The equations relate the properties of a two-dimensional fluid layer uniformly warmed from below and cooled from above. In particular, the equations describe the rate of change of three quantities with respect to time: x is proportional to the rate of convection, y to the horizontal temperature variation, z to the vertical temperature variation; the constants σ, ρ, β are system parameters proportional to the Prandtl number, Rayleigh number, certain physical dimensions of the layer itself.

The Lorenz equations arise in simplified models for lasers, thermosyphons, brushless DC motors, electric circuits, chemical reactions and forward osmosis. From a technical standpoint, the Lorenz system is nonlinear, non-periodic, three-dimensional and deterministic; the Lorenz equations have been the subject of hundreds of research articles, at least one book-length study. One assumes that the parameters σ, ρ, β are positive. Lorenz used the values σ = 10, β = 8 / 3 and ρ = 28; the system exhibits chaotic behavior for these values. If ρ < 1 there is only one equilibrium point, at the origin. This point corresponds to no convection. All orbits converge to the origin, a global attractor, when ρ < 1. A pitchfork bifurcation occurs at ρ = 1, for ρ > 1 two additional critical points appear at: and. These correspond to steady convection; this pair of equilibrium points is stable only if ρ < σ σ + β + 3 σ − β − 1, which can hold only for positive ρ if σ > β + 1. At the critical value, both equilibrium points lose stability through a Hopf bifurcation.

When ρ = 28, σ = 10, β = 8 / 3, the Lorenz system has chaotic solutions. All initial points will tend to an invariant set – the Lorenz attractor – a strange attractor and a fractal, its Hausdorff dimension is estimated to be 2.06 ± 0.01, the correlation dimension is estimated to be 2.05 ± 0.01. The exact Lyapunov dimension formula of the global attractor can be found analytically under classical restrictions on the parameters 3 − 2 σ +

Eugenia Kalnay

Eugenia Kalnay is an Argentine meteorologist and a Distinguished University Professor of Atmospheric and Oceanic Science, part of the University of Maryland College of Computer and Natural Sciences at the University of Maryland, College Park in the United States. She is the recipient of the 54th International Meteorological Organization Prize in 2009 from the World Meteorological Organization for her work on numerical weather prediction, data assimilation, ensemble forecasting; as Director of the Environmental Modeling Center of the National Centers for Environmental Prediction, Kalnay published the 1996 NCEP reanalysis paper, entitled “The NCEP/NCAR 40-year reanalysis project”, one of the most cited papers in the geosciences. She is listed as the author or co-author on over 120 scientific papers and wrote the book Atmospheric Modeling, Data Assimilation and Predictability, published by Cambridge University Press in 2003. Kalnay was born in Argentina and received her undergraduate degree in meteorology from the University of Buenos Aires in 1965.

In 1971, Kalnay became the first woman to receive a PhD in meteorology from MIT, where she was advised by Jule Charney. She became the first female professor in the MIT Department of Meteorology. In 1979 she moved to NASA Goddard and in 1984 became Head of the Global Modeling and Simulation Branch at the Goddard Laboratory for Atmospheres. From 1987 to 1997, Kalnay was the Director of the Environmental Modeling Center of the National Centers for Environmental Prediction, National Weather Service and oversaw the NCEP/NCAR reanalysis project and numerous other projects in data assimilation and ensemble forecasting. After leaving NCEP, Kalnay became the Robert E. Lowry Chair of the School of Meteorology at the University of Oklahoma. In 2002, Kalnay joined the Department of Atmospheric and Oceanic Science at the University of Maryland, College Park and served as chair of the department. Along with James A. Yorke, she co-founded the Weather/Chaos Group at the University of Maryland, which has made discoveries of the local, low-dimensionality of unstable atmospheric regions and the development of the Local Ensemble Kalman filter and Local Ensemble Transform Kalman Filter data assimilation methods.

In addition to the Atmospheric and Ocean Department, Kalnay has appointments in the Institute for Physical Science and Technology and the Center for Computational Science and Mathematical Modeling at the University of Maryland, College Park. In 2008, she was selected as the first Eugenia Brin Endowed Professorship in Data Assimilation. Among the scientific methods Kalnay has pioneered are the breeding method, introduced, along with Zoltan Toth, as a method to identify the growing perturbations in a dynamical system, she was co-author on papers introducing the ensemble methods of Lag Averaged Forecasting and Scaled LAF. In 2017, Kalnay was part of an international team of distinguished scientists who published a study on climate change models in the National Science Review journal; the study argues that there are crucial components missing from current climate models that inform about environmental and economic policies. Kalnay observed that, without including the real feedbacks, predictions for coupled systems cannot work and the model can get away from reality quickly.

Kalnay is a fellow of the American Geophysical Union, the American Meteorological Society, the American Association for the Advancement of Science, the American Academy of Arts and Sciences. She is a member of the National Academy of Engineering, a foreign member of the Academia Europaea, a member of the Argentine National Academy of Physical Sciences. Kalnay has received a number of major awards, including: Lorenz Lecturer, a lecture named after Edward Lorenz and presented at the American Geophysical Union Fall Meeting 54th International Meteorological Organization Prize from the World Meteorological Organization First Eugenia Brin Professorship in Data Assimilation Bjerknes Lecturer, an American Geophysical Union lecture named after Jacob Bjerknes American Meteorological Society Jule G. Charney Award National Aeronautics and Space Administration gold medal for Exceptional Scientific Achievement Academic Webpage

Instability

In numerous fields of study, the component of instability within a system is characterized by some of the outputs or internal states growing without bounds. Not all systems that are not stable are unstable. In structural engineering, a structure can become unstable. Beyond a certain threshold, structural deflections magnify stresses, which in turn increases deflections; this can take the form of crippling. The general field of study is called structural stability. Atmospheric instability is a major component of all weather systems on Earth. In the theory of dynamical systems, a state variable in a system is said to be unstable if it evolves without bounds. A system itself is said to be unstable. In continuous time control theory, a system is unstable if any of the roots of its characteristic equation has real part greater than zero; this is equivalent to any of the eigenvalues of the state matrix having either real part greater than zero, or, for the eigenvalues on the imaginary axis, the algebraic multiplicity being larger than the geometric multiplicity.

The equivalent condition in discrete time is that at least one of the eigenvalues is greater than 1 in absolute value, or that two or more eigenvalues are equal and of unit absolute value. Buckling Elastic instability Drucker stability of a nonlinear constitutive model Biot instability Baroclinic instability Fluid instabilities occur in liquids and plasmas, are characterized by the shape that form. Fluid instabilities include: Ballooning mode instability. Plasma instabilities are categorised into different modes – see this paragraph in plasma stability. Galaxies and star clusters can be unstable, if small perturbations in the gravitational potential cause changes in the density that reinforce the original perturbation; such instabilities require that the motions of stars be correlated, so that the perturbation is not "smeared out" by random motions. After the instability has run its course, the system is "hotter" or rounder than before. Instabilities in stellar systems include: Bar instability of rotating disks Jeans instability Firehose instability Gravothermal instability Radial-orbit instability Various instabilities in cold rotating disks The most common residual disability after any sprain in the body is instability.

Mechanical instability includes insufficient stabilizing structures and mobility that exceed the physiological limits. Functional instability involves a feeling of giving way of the injured joint. Injuries cause impaired postural control in the joint. Individuals with muscular weakness, occult instability, decreased postural control are more susceptible to injury than those with better postural control. Instability leads to an increase in postural sway, the measurement of the time and distance a subject spends away from an ideal center of pressure; the measurement of a subject’s postural sway can be calculated through testing center of pressure, defined as the vertical projection of center of mass on the ground. Investigators have theorized that if injuries to joints cause deafferentation, the interruption of sensory nerve fibers, functional instability a subject’s postural sway should be altered. Joint stability can be enhanced by the use of an external support system, like a brace, to alter body mechanics.

The mechanical support provided by a brace provides cutaneous afferent feedback in maintaining postural control and increasing stability. EFluids Fluid Flow Image Gallery

Mathematical analysis

Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, measure, infinite series, analytic functions. These theories are studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary techniques of analysis. Analysis may be distinguished from geometry. Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, an infinite geometric sum is implicit in Zeno's paradox of the dichotomy. Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids; the explicit use of infinitesimals appears in Archimedes' The Method of Mechanical Theorems, a work rediscovered in the 20th century.

In Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century AD to find the area of a circle. Zu Chongzhi established a method that would be called Cavalieri's principle to find the volume of a sphere in the 5th century; the Indian mathematician Bhāskara II gave examples of the derivative and used what is now known as Rolle's theorem in the 12th century. In the 14th century, Madhava of Sangamagrama developed infinite series expansions, like the power series and the Taylor series, of functions such as sine, cosine and arctangent. Alongside his development of the Taylor series of the trigonometric functions, he estimated the magnitude of the error terms created by truncating these series and gave a rational approximation of an infinite series, his followers at the Kerala School of Astronomy and Mathematics further expanded his works, up to the 16th century. The modern foundations of mathematical analysis were established in 17th century Europe. Descartes and Fermat independently developed analytic geometry, a few decades Newton and Leibniz independently developed infinitesimal calculus, which grew, with the stimulus of applied work that continued through the 18th century, into analysis topics such as the calculus of variations and partial differential equations, Fourier analysis, generating functions.

During this period, calculus techniques were applied to approximate discrete problems by continuous ones. In the 18th century, Euler introduced the notion of mathematical function. Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the modern definition of continuity in 1816, but Bolzano's work did not become known until the 1870s. In 1821, Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra used in earlier work by Euler. Instead, Cauchy formulated calculus in terms of geometric infinitesimals. Thus, his definition of continuity required an infinitesimal change in x to correspond to an infinitesimal change in y, he introduced the concept of the Cauchy sequence, started the formal theory of complex analysis. Poisson, Liouville and others studied partial differential equations and harmonic analysis; the contributions of these mathematicians and others, such as Weierstrass, developed the -definition of limit approach, thus founding the modern field of mathematical analysis.

In the middle of the 19th century Riemann introduced his theory of integration. The last third of the century saw the arithmetization of analysis by Weierstrass, who thought that geometric reasoning was inherently misleading, introduced the "epsilon-delta" definition of limit. Mathematicians started worrying that they were assuming the existence of a continuum of real numbers without proof. Dedekind constructed the real numbers by Dedekind cuts, in which irrational numbers are formally defined, which serve to fill the "gaps" between rational numbers, thereby creating a complete set: the continuum of real numbers, developed by Simon Stevin in terms of decimal expansions. Around that time, the attempts to refine the theorems of Riemann integration led to the study of the "size" of the set of discontinuities of real functions. "monsters" began to be investigated. In this context, Jordan developed his theory of measure, Cantor developed what is now called naive set theory, Baire proved the Baire category theorem.

In the early 20th century, calculus was formalized using an axiomatic set theory. Lebesgue solved the problem of measure, Hilbert introduced Hilbert spaces to solve integral equations; the idea of normed vector space was in the air, in the 1920s Banach created functional analysis. In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. Much of analysis happens in some metric space. Examples of analysis without a metric include functional analysis. Formally, a metric space is an ordered pair where M is a set

Ensemble forecasting

Ensemble forecasting is a method used in numerical weather prediction. Instead of making a single forecast of the most weather, a set of forecasts is produced; this set of forecasts aims to give an indication of the range of possible future states of the atmosphere. Ensemble forecasting is a form of Monte Carlo analysis; the multiple simulations are conducted to account for the two usual sources of uncertainty in forecast models: the errors introduced by the use of imperfect initial conditions, amplified by the chaotic nature of the evolution equations of the atmosphere, referred to as sensitive dependence on initial conditions. Ideally, the verified future atmospheric state should fall within the predicted ensemble spread, the amount of spread should be related to the uncertainty of the forecast. In general, this approach can be used to make probabilistic forecasts of any dynamical system, not just for weather prediction. Today ensemble predictions are made at most of the major operational weather prediction facilities worldwide, including: National Centers for Environmental Prediction European Centre for Medium-Range Weather Forecasts United Kingdom Met Office Météo-France Environment Canada Japan Meteorological Agency Bureau of Meteorology China Meteorological Administration Korea Meteorological Administration CPTEC Ministry of Earth Sciences Experimental ensemble forecasts are made at a number of universities, such as the University of Washington, ensemble forecasts in the US are generated by the US Navy and Air Force.

There are various ways of viewing the data such as spaghetti plots, ensemble means or Postage Stamps where a number of different results from the models run can be compared. As proposed by Edward Lorenz in 1963, it is impossible for long-range forecasts—those made more than two weeks in advance—to predict the state of the atmosphere with any degree of skill owing to the chaotic nature of the fluid dynamics equations involved. Furthermore, existing observation networks have limited spatial and temporal resolution, which introduces uncertainty into the true initial state of the atmosphere. While a set of equations, known as the Liouville equations, exists to determine the initial uncertainty in the model initialization, the equations are too complex to run in real-time with the use of supercomputers; the practical importance of ensemble forecasts derives from the fact that in a chaotic and hence nonlinear system, the rate of growth of forecast error is dependent on starting conditions. An ensemble forecast therefore provides a prior estimate of state-dependent predictability, i.e. an estimate of the types of weather that might occur, given inevitable uncertainties in the forecast initial conditions and in the accuracy of the computational representation of the equations.

These uncertainties limit forecast model accuracy to about six days into the future. The first operational ensemble forecasts were produced for sub-seasonal timescales in 1985. However, it was realised that the philosophy underpinning such forecasts was relevant on shorter timescales – timescales where predictions had been made by purely deterministic means. Edward Epstein recognized in 1969 that the atmosphere could not be described with a single forecast run due to inherent uncertainty, proposed a stochastic dynamic model that produced means and variances for the state of the atmosphere. Although these Monte Carlo simulations showed skill, in 1974 Cecil Leith revealed that they produced adequate forecasts only when the ensemble probability distribution was a representative sample of the probability distribution in the atmosphere, it was not until 1992 that ensemble forecasts began being prepared by the European Centre for Medium-Range Weather Forecasts and the National Centers for Environmental Prediction.

There are two main sources of uncertainty that must be accounted for when making an ensemble weather forecast: initial condition uncertainty and model uncertainty. Initial condition uncertainty arises due to errors in the estimate of the starting conditions for the forecast, both due to limited observations of the atmosphere, uncertainties involved in using indirect measurements, such as satellite data, to measure the state of atmospheric variables. Initial condition uncertainty is represented by perturbing the starting conditions between the different ensemble members; this explores the range of starting conditions consistent with our knowledge of the current state of the atmosphere, together with its past evolution. There are a number of ways to generate these initial condition perturbations; the ECMWF model, the Ensemble Prediction System, uses a combination of singular vectors and an ensemble of data assimilations to simulate the initial probability density. The singular vector perturbations are more active in the extra-tropics, while the EDA perturbations are more active in the tropics.

The NCEP ensemble, the Global Ensemble Forecasting System, uses a technique known as vector breeding. Model uncertainty arises due to the limitations of the forecast model; the process of representing the atmosphere in a computer model involves many simplifications such as the development of parametrisation schemes, which introduce errors into the forecast. Several techniques to represent model uncertainty have been proposed; when developing a parametrisation scheme, many new parameters are introduced to represent simplified physical processes. These parameters may be uncertain