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Riemann Solver
Numerical analysis · Simulation Data analysis · VisualizationPotentials Morse/Long-range potential · Lennard-Jones potential · Yukawa potential · Morse potentialFluid dynamics Finite difference · Finite volume Finite element · Boundary element Lattice Boltzmann · Riemann solver Dissipative particle dynamics Smoothed particle hydrodynamics Turbulence modelsMonte Carlo methods Integration · Gibbs sampling · Metropolis algorithmParticle N-body · Particle-in-cell Molecular dynamicsScientists Godunov · Ulam · von Neumann · Galerkin · Lorenz · Wilsonv t eA Riemann solver
Riemann solver
is a numerical method used to solve a Riemann problem
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N-body Simulation
In physics and astronomy, an N-body simulation
N-body simulation
is a simulation of a dynamical system of particles, usually under the influence of physical forces, such as gravity (see n-body problem). N-body simulations are widely used tools in astrophysics, from investigating the dynamics of few-body systems like the Earth-Moon- Sun
Sun
system to understanding the evolution of the large-scale structure of the universe.[1] In physical cosmology, N-body simulations are used to study processes of non-linear structure formation such as galaxy filaments and galaxy halos from the influence of dark matter
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Kenneth G. Wilson
Kenneth Geddes "Ken" Wilson (June 8, 1936 – June 15, 2013) was an American theoretical physicist and a pioneer in leveraging computers for studying particle physics. He was awarded the 1982 Nobel Prize in Physics for his work on phase transitions—illuminating the subtle essence of phenomena like melting ice and emerging magnetism. It was embodied in his fundamental work on the renormalization group.Contents1 Life 2 Work 3 Awards and honors 4 See also 5 Notes 6 External linksLife[edit] Wilson was born on June 8, 1936, in Waltham, Massachusetts, the oldest child of Emily Buckingham Wilson and E. Bright Wilson, a prominent chemist at Harvard University, who did important work on microwave emissions. His mother also trained as a physicist
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Molecular Dynamics
Molecular dynamics
Molecular dynamics
(MD) is a computer simulation method for studying the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic evolution of the system. In the most common version, the trajectories of atoms and molecules are determined by numerically solving Newton's equations of motion for a system of interacting particles, where forces between the particles and their potential energies are often calculated using interatomic potentials or molecular mechanics force fields
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John Von Neumann
John von Neumann
John von Neumann
(/vɒn ˈnɔɪmən/; Hungarian: Neumann János Lajos, pronounced [ˈnɒjmɒn ˈjaːnoʃ ˈlɒjoʃ]; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, and polymath
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Numerical Analysis
Numerical analysis
Numerical analysis
is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). One of the earliest mathematical writings is a Babylonian tablet from the Yale Babylonian Collection
Yale Babylonian Collection
(YBC 7289), which gives a sexagesimal numerical approximation of the square root of 2, the length of the diagonal in a unit square. Being able to compute the sides of a triangle (and hence, being able to compute square roots) is extremely important, for instance, in astronomy, carpentry and construction.[2] Numerical analysis
Numerical analysis
continues this long tradition of practical mathematical calculations
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Metropolis–Hastings Algorithm
In statistics and in statistical physics, the Metropolis–Hastings algorithm is a Markov chain Monte Carlo
Markov chain Monte Carlo
(MCMC) method for obtaining a sequence of random samples from a probability distribution for which direct sampling is difficult. This sequence can be used to approximate the distribution (e.g., to generate a histogram), or to compute an integral (such as an expected value). Metropolis–Hastings and other MCMC algorithms are generally used for sampling from multi-dimensional distributions, especially when the number of dimensions is high. For single-dimensional distributions, other methods are usually available (e.g
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Gibbs Sampling
In statistics, Gibbs sampling or a Gibbs sampler is a Markov chain Monte Carlo (MCMC) algorithm for obtaining a sequence of observations which are approximated from a specified multivariate probability distribution, when direct sampling is difficult. This sequence can be used to approximate the joint distribution (e.g., to generate a histogram of the distribution); to approximate the marginal distribution of one of the variables, or some subset of the variables (for example, the unknown parameters or latent variables); or to compute an integral (such as the expected value of one of the variables). Typically, some of the variables correspond to observations whose values are known, and hence do not need to be sampled. Gibbs sampling is commonly used as a means of statistical inference, especially Bayesian inference. It is a randomized algorithm (i.e
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Monte Carlo Integration
In mathematics, Monte Carlo integration
Monte Carlo integration
is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that numerically computes a definite integral. While other algorithms usually evaluate the integrand at a regular grid,[1] Monte Carlo randomly choose points at which the integrand is evaluated.[2] This method is particularly useful for higher-dimensional integrals.[3] There are different methods to perform a Monte Carlo integration, such as uniform sampling, stratified sampling, importance sampling, sequential Monte Carlo (a.k.a
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Boris Galerkin
Boris Grigoryevich Galerkin (Russian: Бори́с Григо́рьевич Галёркин, surname more accurately romanized as Galyorkin; 4 March [O.S. 20 February] 1871 – 12 July 1945), born in Polotsk, Vitebsk Governorate, Russian Empire, was a Soviet mathematician and an engineer.Contents1 Biography1.1 Early days 1.2 Political activities and imprisonment 1.3 Academic 1.4 War times and death2 Mathematical contributions 3 References 4 External linksBiography[edit] Early days[edit] Galerkin was born on March 4 [O.S. February 20, 1871] 1871 in Polotsk, Vitebsk Governorate, Russian Empire, now part of Belarus, to Jewish[1][2][3] parents, Girsh-Shleym(Hirsh-Shleym) Galerkin and Perla Basia Galerkina. His parents owned a house in the town, but the homecraft they made did not bring enough money, so at the age of 12, Boris started working as calligrapher in the court
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Turbulence Modeling
Turbulence
Turbulence
modeling is the construction and use of a model to predict the effects of turbulence. A turbulent fluid flow has features on many different length scales, which all interact with each other. A common approach is to average the governing equations of the flow, in order to focus on large-scale and non-fluctuating features of the flow. However, the effects of the small scales and fluctuating parts must be modelled.[1]Contents1 Closure problem 2 Eddy viscosity 3 Prandtl's mixing-length concept 4 Smagorinsky model for the sub-grid scale eddy viscosity 5 Spalart–Allmaras, k–ε and k–ω models 6 Common models 7 References7.1 Notes 7.2 OtherClosure problem[edit] The Navier–Stokes equations
Navier–Stokes equations
govern the velocity and pressure of a fluid flow. In a turbulent flow, each of these quantities may be decomposed into a mean part and a fluctuating part
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Edward Norton Lorenz
Edward Norton
Edward Norton
Lorenz (May 23, 1917 – April 16, 2008)[1][2] was an American mathematician, meteorologist, and a pioneer of chaos theory, along with Mary Cartwright.[3] He introduced the strange attractor notion and coined the term butterfly effect.Contents1 Biography 2 Awards 3 Work 4 Publications 5 See also 6 References 7 External linksBiography[edit] Lorenz was born in West Hartford, Connecticut.[4] He studied mathematics at both Dartmouth College
Dartmouth College
in New Hampshire
New Hampshire
and Harvard University in Cambridge, Massachusetts. From 1942 until 1946, he served as a meteorologist for the United States Army Air Corps. After his return from World War II, he decided to study meteorology.[2] Lorenz earned two degrees in the area from the Massachusetts Institute of Technology where he later was a professor for many years
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Dissipative Particle Dynamics
Dissipative particle dynamics (DPD) is a stochastic simulation technique for simulating the dynamic and rheological properties of simple and complex fluids. It was initially devised by Hoogerbrugge and Koelman[1][2] to avoid the lattice artifacts of the so-called lattice gas automata and to tackle hydrodynamic time and space scales beyond those available with molecular dynamics (MD). It was subsequently reformulated and slightly modified by P. Español[3] to ensure the proper thermal equilibrium state
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Lattice Boltzmann Methods
Lattice Boltzmann methods (LBM) (or thermal Lattice Boltzmann methods (TLBM)) is a class of computational fluid dynamics (CFD) methods for fluid simulation. Instead of solving the Navier–Stokes equations, the discrete Boltzmann equation
Boltzmann equation
is solved to simulate the flow of a Newtonian fluid with collision models such as Bhatnagar–Gross–Krook (BGK)
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Numerical Method
In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm.Contents1 Mathematical definition 2 Consistency 3 Convergence 4 ReferencesMathematical definition[edit] Let F ( x , y ) = 0 displaystyle F(x,y)=0 be a well-posed problem, i.e
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Stanislaw Ulam
Stanisław Marcin Ulam (pronounced ['staɲiswaf 'mart͡ɕin 'ulam]; 13 April 1909 – 13 May 1984) was a Polish-American scientist in the fields of mathematics and nuclear physics. He participated in the Manhattan Project, originated the Teller–Ulam design
Teller–Ulam design
of thermonuclear weapons, discovered the concept of cellular automaton, invented the Monte Carlo method of computation, and suggested nuclear pulse propulsion. In pure and applied mathematics, he proved some theorems and proposed several conjectures. Born into a wealthy Polish Jewish
Polish Jewish
family, Ulam studied mathematics at the Lwów Polytechnic Institute, where he earned his PhD
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
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