In numerical analysis, adaptive mesh refinement (AMR) is a method of adapting the accuracy of a solution within certain sensitive or turbulent regions of simulation, dynamically and during the time the solution is being calculated. When solutions are calculated numerically, they are often limited to pre-determined quantified grids as in the Cartesian plane which constitute the computational grid, or 'mesh'. Many problems in numerical analysis, however, do not require a uniform precision in the numerical grids used for graph plotting or computational simulation, and would be better suited if specific areas of graphs which needed precision could be refined in quantification only in the regions requiring the added precision. Adaptive mesh refinement provides such a dynamic programming environment for adapting the precision of the numerical computation based on the requirements of a computation problem in specific areas of multi-dimensional graphs which need precision while leaving the other regions of the multi-dimensional graphs at lower levels of precision and resolution.
Auxiliary-field Monte Carlo is a method that allows the calculation, by use of Monte Carlo techniques, of averages of operators in many-body quantum mechanical or classical problems.
High-resolution schemes are used in the numerical solution of partial differential equations where high accuracy is required in the presence of shocks or discontinuities. They have the following properties:
Bram van Leer is Arthur B. Modine Emeritus Professor of aerospace engineering at the University of Michigan, in Ann Arbor. He specializes in Computational fluid dynamics (CFD), fluid dynamics, and numerical analysis. His most influential work lies in CFD, a field he helped modernize from 1970 onwards. An appraisal of his early work has been given by C. Hirsch (1979)
Alexandre Joel Chorin is an American mathematician known for his contributions to computational fluid mechanics, turbulence, and computational statistical mechanics.
A Riemann solver is a numerical method used to solve a Riemann problem. They are heavily used in computational fluid dynamics and computational magnetohydrodynamics.
In physics, Gauss's law for magnetism is one of the four Maxwell's equations that underlie classical electrodynamics. It states that the magnetic field B has divergence equal to zero, in other words, that it is a solenoidal vector field. It is equivalent to the statement that magnetic monopoles do not exist. Rather than "magnetic charges", the basic entity for magnetism is the magnetic dipole.
In computational fluid dynamics, the volume of fluid (VOF) method is a free-surface modelling technique, i.e. a numerical technique for tracking and locating the free surface. It belongs to the class of Eulerian methods which are characterized by a mesh that is either stationary or is moving in a certain prescribed manner to accommodate the evolving shape of the interface. As such, VOF is an advection scheme—a numerical recipe that allows the programmer to track the shape and position of the interface, but it is not a standalone flow solving algorithm. The Navier–Stokes equations describing the motion of the flow have to be solved separately. The same applies for all other advection algorithms.
In computational fluid dynamics, the immersed boundary method originally referred to an approach developed by Charles Peskin in 1972 to simulate fluid-structure (fiber) interactions. Treating the coupling of the structure deformations and the fluid flow poses a number of challenging problems for numerical simulations. In the immersed boundary method the fluid is represented on an Eulerian coordinate and the structure is represented on a Lagrangian coordinate. For Newtonian fluids governed by the incompressible Navier–Stokes equations, the fluid equations are
Amiram Harten was an American/Israeli applied mathematician. Harten made fundamental contribution to the development of high-resolution schemes for the solution of hyperbolic partial differential equations. Among other contributions, he developed the total variation diminishing scheme, which gives an oscillation free solution for flow with shocks.
Burton Wendroff is an American applied mathematician known for his contributions to the development of numerical methods for the solution of hyperbolic partial differential equations. The Lax–Wendroff method for the solution of hyperbolic PDE is named for Wendroff.
Francis Harvey Harlow was an American theoretical physicist known for his work in the field of fluid dynamics. He was a researcher at Los Alamos National Laboratory, Los Alamos, New Mexico. Harlow is credited with establishing the science of computational fluid dynamics (CFD) as an important discipline.
John B. Bell is an American mathematician and the Chief Scientist of the Computational Research Division at the Lawrence Berkeley National Laboratory. He has made contributions in the areas of finite difference methods, numerical methods for low Mach number flows, adaptive mesh refinement, interface tracking and parallel computing. He has also worked on the application of these numerical methods to problems from a broad range of fields, including combustion, shock physics, seismology, flow in porous media and astrophysics.
The following is a timeline of scientific computing, also known as computational science.
The following is a timeline of numerical analysis after 1945, and deals with developments after the invention of the modern electronic computer, which began during Second World War. For a fuller history of the subject before this period, see timeline and history of mathematics.
Dimitris Drikakis, PhD, FRAeS, CEng, is a Greek-British applied scientist, engineer and university professor. His research is multidisciplinary. It covers fluid dynamics, computational fluid dynamics, acoustics, heat transfer, computational science from molecular to macro scale, materials, machine learning, and emerging technologies. He has applied his research to diverse fields such as Aerospace & Defence, Biomedical, and Energy and Environment Sectors. He received The William Penney Fellowship Award by the Atomic Weapons Establishment to recognise his contributions to compressible fluid dynamics. He was also the winner of NEF's Innovator of the Year Award by the UK's Institute of Innovation and Knowledge Exchange for a new generation carbon capture nanotechnology that uses carbon nanotubes for filtering out carbon dioxide and other gases.
Joseph E. Oliger was an American computer scientist and professor at Stanford University. Oliger was the co-founder of the Science in Computational and Mathematical Engineering degree program at Stanford, and served as the director of the Research Institute for Advanced Computer Science.
Integrable algorithms are numerical algorithms that rely on basic ideas from the mathematical theory of integrable systems.
James Thomas Beale is an American mathematician, specializing in fluid dynamics, partial differential equations, and numerical analysis.
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