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.
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 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.
Potential flowIn fluid dynamics, potential flow (or ideal flow) describes the velocity field as the gradient of a scalar function: the velocity potential. As a result, a potential flow is characterized by an irrotational velocity field, which is a valid approximation for several applications. The irrotationality of a potential flow is due to the curl of the gradient of a scalar always being equal to zero. In the case of an incompressible flow the velocity potential satisfies Laplace's equation, and potential theory is applicable.
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.
Groundwater flow equationUsed in hydrogeology, the groundwater flow equation is the mathematical relationship which is used to describe the flow of groundwater through an aquifer. The transient flow of groundwater is described by a form of the diffusion equation, similar to that used in heat transfer to describe the flow of heat in a solid (heat conduction). The steady-state flow of groundwater is described by a form of the Laplace equation, which is a form of potential flow and has analogs in numerous fields.
Groundwater modelGroundwater models are computer models of groundwater flow systems, and are used by hydrologists and hydrogeologists. Groundwater models are used to simulate and predict aquifer conditions. An unambiguous definition of "groundwater model" is difficult to give, but there are many common characteristics. A groundwater model may be a scale model or an electric model of a groundwater situation or aquifer. Groundwater models are used to represent the natural groundwater flow in the environment.
Hydrogen economyThe hydrogen economy uses hydrogen to decarbonize economic sectors which are hard to electrify, essentially, the "hard-to-abate" sectors such as cement, steel, long-haul transport, etc. In order to phase out fossil fuels and limit climate change, hydrogen can be created from water using renewable sources such as wind and solar, and its combustion only releases water vapor into the atmosphere. Although with a very low volumetric energy density hydrogen is an energetic fuel, frequently used as rocket fuel, but numerous technical challenges prevent the creation of a large-scale hydrogen economy.
Hydrogen infrastructureA hydrogen infrastructure is the infrastructure of hydrogen pipeline transport, points of hydrogen production and hydrogen stations (sometimes clustered as a hydrogen highway) for distribution as well as the sale of hydrogen fuel, and thus a crucial prerequisite before a successful commercialization of automotive fuel cell technology. A hydrogen highway is a chain of hydrogen-equipped filling stations and other infrastructure along a road or highway which allow hydrogen vehicles to travel.
Ion exchangeIon exchange is a reversible interchange of one kind of ion present in an insoluble solid with another of like charge present in a solution surrounding the solid with the reaction being used especially for softening or making water demineralised, the purification of chemicals and separation of substances. Ion exchange usually describes a process of purification of aqueous solutions using solid polymeric ion-exchange resin. More precisely, the term encompasses a large variety of processes where ions are exchanged between two electrolytes.
Hydrogen storageSeveral methods exist for storing hydrogen. These include mechanical approaches such as using high pressures and low temperatures, or employing chemical compounds that release H2 upon demand. While large amounts of hydrogen are produced by various industries, it is mostly consumed at the site of production, notably for the synthesis of ammonia. For many years hydrogen has been stored as compressed gas or cryogenic liquid, and transported as such in cylinders, tubes, and cryogenic tanks for use in industry or as propellant in space programs.
Partial differential equationIn mathematics, a partial differential equation (PDE) is an equation which computes a function between various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similar to how x is thought of as an unknown number to be solved for in an algebraic equation like x2 − 3x + 2 = 0. However, it is usually impossible to write down explicit formulas for solutions of partial differential equations.
Hydrogen vehicleA hydrogen vehicle is a vehicle that uses hydrogen fuel for motive power. Hydrogen vehicles include hydrogen-fueled space rockets, as well as ships and aircraft. Power is generated by converting the chemical energy of hydrogen to mechanical energy, either by reacting hydrogen with oxygen in a fuel cell to power electric motors or, less commonly, by burning hydrogen in an internal combustion engine. there are two models of hydrogen cars publicly available in select markets: the Toyota Mirai (2014–), which is the world's first mass-produced dedicated fuel cell electric vehicle (FCEV), and the Hyundai Nexo (2018–).
Double layer (surface science)In surface science, a double layer (DL, also called an electrical double layer, EDL) is a structure that appears on the surface of an object when it is exposed to a fluid. The object might be a solid particle, a gas bubble, a liquid droplet, or a porous body. The DL refers to two parallel layers of charge surrounding the object. The first layer, the surface charge (either positive or negative), consists of ions which are adsorbed onto the object due to chemical interactions.
Open-channel flowIn fluid mechanics and hydraulics, open-channel flow is a type of liquid flow within a conduit with a free surface, known as a channel. The other type of flow within a conduit is pipe flow. These two types of flow are similar in many ways but differ in one important respect: open-channel flow has a free surface, whereas pipe flow does not. Open-channel flow can be classified and described in various ways based on the change in flow depth with respect to time and space.
Ion-exchange resinAn ion-exchange resin or ion-exchange polymer is a resin or polymer that acts as a medium for ion exchange. It is an insoluble matrix (or support structure) normally in the form of small (0.25–1.43 mm radius) microbeads, usually white or yellowish, fabricated from an organic polymer substrate. The beads are typically porous (with a specific size distribution that will affect its properties), providing a large surface area on and inside them where the trapping of ions occurs along with the accompanying release of other ions, and thus the process is called ion exchange.
Laminar flowIn fluid dynamics, laminar flow (ˈlæmənər) is characterized by fluid particles following smooth paths in layers, with each layer moving smoothly past the adjacent layers with little or no mixing. At low velocities, the fluid tends to flow without lateral mixing, and adjacent layers slide past one another like playing cards. There are no cross-currents perpendicular to the direction of flow, nor eddies or swirls of fluids. In laminar flow, the motion of the particles of the fluid is very orderly with particles close to a solid surface moving in straight lines parallel to that surface.
Chemical reactionA chemical reaction is a process that leads to the chemical transformation of one set of chemical substances to another. Classically, chemical reactions encompass changes that only involve the positions of electrons in the forming and breaking of chemical bonds between atoms, with no change to the nuclei (no change to the elements present), and can often be described by a chemical equation. Nuclear chemistry is a sub-discipline of chemistry that involves the chemical reactions of unstable and radioactive elements where both electronic and nuclear changes can occur.
