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 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.
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 linear algebraNumerical linear algebra, sometimes called applied linear algebra, is the study of how matrix operations can be used to create computer algorithms which efficiently and accurately provide approximate answers to questions in continuous mathematics. It is a subfield of numerical analysis, and a type of linear algebra. Computers use floating-point arithmetic and cannot exactly represent irrational data, so when a computer algorithm is applied to a matrix of data, it can sometimes increase the difference between a number stored in the computer and the true number that it is an approximation of.
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.
Euler methodIn mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit method for numerical integration of ordinary differential equations and is the simplest Runge–Kutta method. The Euler method is named after Leonhard Euler, who first proposed it in his book Institutionum calculi integralis (published 1768–1870).
Explicit and implicit methodsExplicit and implicit methods are approaches used in numerical analysis for obtaining numerical approximations to the solutions of time-dependent ordinary and partial differential equations, as is required in computer simulations of physical processes. Explicit methods calculate the state of a system at a later time from the state of the system at the current time, while implicit methods find a solution by solving an equation involving both the current state of the system and the later one.
Verlet integrationVerlet integration (vɛʁˈlɛ) is a numerical method used to integrate Newton's equations of motion. It is frequently used to calculate trajectories of particles in molecular dynamics simulations and computer graphics. The algorithm was first used in 1791 by Jean Baptiste Delambre and has been rediscovered many times since then, most recently by Loup Verlet in the 1960s for use in molecular dynamics. It was also used by P. H. Cowell and A. C. C.
Symplectic integratorIn mathematics, a symplectic integrator (SI) is a numerical integration scheme for Hamiltonian systems. Symplectic integrators form the subclass of geometric integrators which, by definition, are canonical transformations. They are widely used in nonlinear dynamics, molecular dynamics, discrete element methods, accelerator physics, plasma physics, quantum physics, and celestial mechanics. Symplectic integrators are designed for the numerical solution of Hamilton's equations, which read where denotes the position coordinates, the momentum coordinates, and is the Hamiltonian.
High frequencyHigh frequency (HF) is the ITU designation for the range of radio frequency electromagnetic waves (radio waves) between 3 and 30 megahertz (MHz). It is also known as the decameter band or decameter wave as its wavelengths range from one to ten decameters (ten to one hundred meters). Frequencies immediately below HF are denoted medium frequency (MF), while the next band of higher frequencies is known as the very high frequency (VHF) band. The HF band is a major part of the shortwave band of frequencies, so communication at these frequencies is often called shortwave radio.
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.
Ultra high frequencyUltra high frequency (UHF) is the ITU designation for radio frequencies in the range between 300 megahertz (MHz) and 3 gigahertz (GHz), also known as the decimetre band as the wavelengths range from one meter to one tenth of a meter (one decimeter). Radio waves with frequencies above the UHF band fall into the super-high frequency (SHF) or microwave frequency range. Lower frequency signals fall into the VHF (very high frequency) or lower bands.
Super high frequencySuper high frequency (SHF) is the ITU designation for radio frequencies (RF) in the range between 3 and 30 gigahertz (GHz). This band of frequencies is also known as the centimetre band or centimetre wave as the wavelengths range from one to ten centimetres. These frequencies fall within the microwave band, so radio waves with these frequencies are called microwaves. The small wavelength of microwaves allows them to be directed in narrow beams by aperture antennas such as parabolic dishes and horn antennas, so they are used for point-to-point communication and data links and for radar.
Molecular dynamicsMolecular dynamics (MD) is a computer simulation method for analyzing 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 mechanical force fields.
Stiff equationIn mathematics, a stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable, unless the step size is taken to be extremely small. It has proven difficult to formulate a precise definition of stiffness, but the main idea is that the equation includes some terms that can lead to rapid variation in the solution.
Structural biologyStructural biology is a field that is many centuries old which, as defined by the Journal of Structural Biology, deals with structural analysis of living material (formed, composed of, and/or maintained and refined by living cells) at every level of organization. Early structural biologists throughout the 19th and early 20th centuries were primarily only able to study structures to the limit of the naked eye's visual acuity and through magnifying glasses and light microscopes.
Computational electromagneticsComputational electromagnetics (CEM), computational electrodynamics or electromagnetic modeling is the process of modeling the interaction of electromagnetic fields with physical objects and the environment. It typically involves using computer programs to compute approximate solutions to Maxwell's equations to calculate antenna performance, electromagnetic compatibility, radar cross section and electromagnetic wave propagation when not in free space.
Radio frequencyRadio frequency (RF) is the oscillation rate of an alternating electric current or voltage or of a magnetic, electric or electromagnetic field or mechanical system in the frequency range from around 20kHz to around 300GHz. This is roughly between the upper limit of audio frequencies and the lower limit of infrared frequencies. These are the frequencies at which energy from an oscillating current can radiate off a conductor into space as radio waves, so they are used in radio technology, among other uses.
Numerical cognitionNumerical cognition is a subdiscipline of cognitive science that studies the cognitive, developmental and neural bases of numbers and mathematics. As with many cognitive science endeavors, this is a highly interdisciplinary topic, and includes researchers in cognitive psychology, developmental psychology, neuroscience and cognitive linguistics. This discipline, although it may interact with questions in the philosophy of mathematics, is primarily concerned with empirical questions.
Structural bioinformaticsStructural bioinformatics is the branch of bioinformatics that is related to the analysis and prediction of the three-dimensional structure of biological macromolecules such as proteins, RNA, and DNA. It deals with generalizations about macromolecular 3D structures such as comparisons of overall folds and local motifs, principles of molecular folding, evolution, binding interactions, and structure/function relationships, working both from experimentally solved structures and from computational models.
