Numerical analysisNumerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts.
Numerical methods for ordinary differential equationsNumerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals. Many differential equations cannot be solved exactly. For practical purposes, however – such as in engineering – a numeric approximation to the solution is often sufficient. The algorithms studied here can be used to compute such an approximation.
Numerical methods for partial differential equationsNumerical methods for partial differential equations is the branch of numerical analysis that studies the numerical solution of partial differential equations (PDEs). In principle, specialized methods for hyperbolic, parabolic or elliptic partial differential equations exist. Finite difference method In this method, functions are represented by their values at certain grid points and derivatives are approximated through differences in these values.
Numerical stabilityIn the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation. In numerical linear algebra, the principal concern is instabilities caused by proximity to singularities of various kinds, such as very small or nearly colliding eigenvalues.
Molten salt reactorA molten salt reactor (MSR) is a class of nuclear fission reactor in which the primary nuclear reactor coolant and/or the fuel is a mixture of molten salt with a fissionable material. Two research MSRs operated in the United States in the mid-20th century. The 1950s Aircraft Reactor Experiment (ARE) was primarily motivated by the technology's compact size, while the 1960s Molten-Salt Reactor Experiment (MSRE) aimed to demonstrate a nuclear power plant using a thorium fuel cycle in a breeder reactor.
Numerical integrationIn analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of definite integrals. The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for numerical integration, especially as applied to one-dimensional integrals.
Small modular reactorSmall modular reactors (SMRs) are a proposed class of nuclear fission reactors, smaller than conventional nuclear reactors, which can be built in one location (such as a factory), then shipped, commissioned, and operated at a separate site. The term SMR refers to the size, capacity and modular construction only, not to the reactor type and the nuclear process which is applied. Designs range from scaled down versions of existing designs to generation IV designs.
Numerical methods for linear least squaresNumerical methods for linear least squares entails the numerical analysis of linear least squares problems. A general approach to the least squares problem can be described as follows. Suppose that we can find an n by m matrix S such that XS is an orthogonal projection onto the image of X. Then a solution to our minimization problem is given by simply because is exactly a sought for orthogonal projection of onto an image of X (see the picture below and note that as explained in the next section the image of X is just a subspace generated by column vectors of X).
Supercritical water reactorThe supercritical water reactor (SCWR) is a concept Generation IV reactor, designed as a light water reactor (LWR) that operates at supercritical pressure (i.e. greater than 22.1 MPa). The term critical in this context refers to the critical point of water, and must not be confused with the concept of criticality of the nuclear reactor. The water heated in the reactor core becomes a supercritical fluid above the critical temperature of 374 °C, transitioning from a fluid more resembling liquid water to a fluid more resembling saturated steam (which can be used in a steam turbine), without going through the distinct phase transition of boiling.
Eigenvalue algorithmIn numerical analysis, one of the most important problems is designing efficient and stable algorithms for finding the eigenvalues of a matrix. These eigenvalue algorithms may also find eigenvectors. Eigenvalues and eigenvectors and Generalized eigenvector Given an n × n square matrix A of real or complex numbers, an eigenvalue λ and its associated generalized eigenvector v are a pair obeying the relation where v is a nonzero n × 1 column vector, I is the n × n identity matrix, k is a positive integer, and both λ and v are allowed to be complex even when A is real.
Eigenvalues and eigenvectorsIn linear algebra, an eigenvector (ˈaɪgənˌvɛktər) or characteristic vector of a linear transformation is a nonzero vector that changes at most by a constant factor when that linear transformation is applied to it. The corresponding eigenvalue, often represented by , is the multiplying factor. Geometrically, a transformation matrix rotates, stretches, or shears the vectors it acts upon. The eigenvectors for a linear transformation matrix are the set of vectors that are only stretched, with no rotation or shear.
Boiling water reactorA boiling water reactor (BWR) is a type of light water nuclear reactor used for the generation of electrical power. It is the second most common type of electricity-generating nuclear reactor after the pressurized water reactor (PWR), which is also a type of light water nuclear reactor. The main difference between a BWR and PWR is that in a BWR, the reactor core heats water, which turns to steam and then drives a steam turbine. In a PWR, the reactor core heats water, which does not boil.
Molten-Salt Reactor ExperimentThe Molten-Salt Reactor Experiment (MSRE) was an experimental molten salt reactor research reactor at the Oak Ridge National Laboratory (ORNL). This technology was researched through the 1960s, the reactor was constructed by 1964, it went critical in 1965, and was operated until 1969. The costs of a cleanup project were estimated at about $130 million. The MSRE was a 7.4 MWth test reactor simulating the neutronic "kernel" of a type of inherently safer epithermal thorium breeder reactor called the liquid fluoride thorium reactor.
Reactor-grade plutoniumReactor-grade plutonium (RGPu) is the isotopic grade of plutonium that is found in spent nuclear fuel after the uranium-235 primary fuel that a nuclear power reactor uses has burnt up. The uranium-238 from which most of the plutonium isotopes derive by neutron capture is found along with the U-235 in the low enriched uranium fuel of civilian reactors.
Gaseous diffusionGaseous diffusion is a technology that was used to produce enriched uranium by forcing gaseous uranium hexafluoride (UF6) through microporous membranes. This produces a slight separation (enrichment factor 1.0043) between the molecules containing uranium-235 (235U) and uranium-238 (238U). By use of a large cascade of many stages, high separations can be achieved. It was the first process to be developed that was capable of producing enriched uranium in industrially useful quantities, but is nowadays considered obsolete, having been superseded by the more-efficient gas centrifuge process.
Eigendecomposition of a matrixIn linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matrix being factorized is a normal or real symmetric matrix, the decomposition is called "spectral decomposition", derived from the spectral theorem. Eigenvalue, eigenvector and eigenspace A (nonzero) vector v of dimension N is an eigenvector of a square N × N matrix A if it satisfies a linear equation of the form for some scalar λ.
Enriched uraniumEnriched uranium is a type of uranium in which the percent composition of uranium-235 (written 235U) has been increased through the process of isotope separation. Naturally occurring uranium is composed of three major isotopes: uranium-238 (238U with 99.2739–99.2752% natural abundance), uranium-235 (235U, 0.7198–0.7202%), and uranium-234 (234U, 0.0050–0.0059%). 235U is the only nuclide existing in nature (in any appreciable amount) that is fissile with thermal neutrons.
NeutronThe neutron is a subatomic particle, symbol _Neutron or _Neutron0, which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons behave similarly within the nucleus, and each has a mass of approximately one dalton, they are both referred to as nucleons. Their properties and interactions are described by nuclear physics. Protons and neutrons are not elementary particles; each is composed of three quarks.
Nuclear reactorA nuclear reactor is a device used to initiate and control a fission nuclear chain reaction or nuclear fusion reactions. Nuclear reactors are used at nuclear power plants for electricity generation and in nuclear marine propulsion. Heat from nuclear fission is passed to a working fluid (water or gas), which in turn runs through steam turbines. These either drive a ship's propellers or turn electrical generators' shafts. Nuclear generated steam in principle can be used for industrial process heat or for district heating.
Fast-neutron reactorA fast-neutron reactor (FNR) or fast-spectrum reactor or simply a fast reactor is a category of nuclear reactor in which the fission chain reaction is sustained by fast neutrons (carrying energies above 1 MeV or greater, on average), as opposed to slow thermal neutrons used in thermal-neutron reactors. Such a fast reactor needs no neutron moderator, but requires fuel that is relatively rich in fissile material when compared to that required for a thermal-neutron reactor.
