DiscretizationIn applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers. Dichotomization is the special case of discretization in which the number of discrete classes is 2, which can approximate a continuous variable as a binary variable (creating a dichotomy for modeling purposes, as in binary classification).
Finite element methodThe finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. The FEM is a general numerical method for solving partial differential equations in two or three space variables (i.e., some boundary value problems).
Finite difference methodIn numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time interval (if applicable) are discretized, or broken into a finite number of steps, and the value of the solution at these discrete points is approximated by solving algebraic equations containing finite differences and values from nearby points.
Finite differenceA finite difference is a mathematical expression of the form f (x + b) − f (x + a). If a finite difference is divided by b − a, one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems. The difference operator, commonly denoted is the operator that maps a function f to the function defined by A difference equation is a functional equation that involves the finite difference operator in the same way as a differential equation involves derivatives.
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.
Earth's crustEarth's crust is Earth's thick outer shell of rock, referring to less than 1% of the Earth's radius and volume. It is the top component of the lithosphere, a division of Earth's layers that includes the crust and the upper part of the mantle. The lithosphere is broken into tectonic plates whose motion allows heat to escape from the interior of the Earth into space. The crust lies on top of the mantle, a configuration that is stable because the upper mantle is made of peridotite and is therefore significantly denser than the crust.
Discrete mathematicsDiscrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous functions). Objects studied in discrete mathematics include integers, graphs, and statements in logic. By contrast, discrete mathematics excludes topics in "continuous mathematics" such as real numbers, calculus or Euclidean geometry.
Continental crustContinental crust is the layer of igneous, metamorphic, and sedimentary rocks that forms the geological continents and the areas of shallow seabed close to their shores, known as continental shelves. This layer is sometimes called sial because its bulk composition is richer in aluminium silicates (Al-Si) and has a lower density compared to the oceanic crust, called sima which is richer in magnesium silicate (Mg-Si) minerals.
Oceanic crustOceanic crust is the uppermost layer of the oceanic portion of the tectonic plates. It is composed of the upper oceanic crust, with pillow lavas and a dike complex, and the lower oceanic crust, composed of troctolite, gabbro and ultramafic cumulates. The crust overlies the rigid uppermost layer of the mantle. The crust and the rigid upper mantle layer together constitute oceanic lithosphere. Oceanic crust is primarily composed of mafic rocks, or sima, which is rich in iron and magnesium.
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.
Upper mantleThe upper mantle of Earth is a very thick layer of rock inside the planet, which begins just beneath the crust (at about under the oceans and about under the continents) and ends at the top of the lower mantle at . Temperatures range from approximately at the upper boundary with the crust to approximately at the boundary with the lower mantle. Upper mantle material that has come up onto the surface comprises about 55% olivine, 35% pyroxene, and 5 to 10% of calcium oxide and aluminum oxide minerals such as plagioclase, spinel, or garnet, depending upon depth.
Earth's mantleEarth's mantle is a layer of silicate rock between the crust and the outer core. It has a mass of and thus makes up 67% of the mass of Earth. It has a thickness of making up about 84% of Earth's volume. It is predominantly solid but, on geologic time scales, it behaves as a viscous fluid, sometimes described as having the consistency of caramel. Partial melting of the mantle at mid-ocean ridges produces oceanic crust, and partial melting of the mantle at subduction zones produces continental crust.
Crust (geology)In geology, the crust is the outermost solid shell of a rocky planet, dwarf planet, or natural satellite. It is usually distinguished from the underlying mantle by its chemical makeup; however, in the case of icy satellites, it may be distinguished based on its phase (solid crust vs. liquid mantle). The crusts of Earth, Mercury, Venus, Mars, Io, the Moon and other planetary bodies formed via igneous processes and were later modified by erosion, impact cratering, volcanism, and sedimentation.
Sheeted dyke complexA sheeted dyke complex, or sheeted dike complex, is a series of sub-parallel intrusions of igneous rock, forming a layer within the oceanic crust. At mid-ocean ridges, dykes are formed when magma beneath areas of tectonic plate divergence travels through a fracture in the earlier formed oceanic crust, feeding the lavas above and cooling below the seafloor forming upright columns of igneous rock. Magma continues to cool, as the existing seafloor moves away from the area of divergence, and additional magma is intruded and cools.
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.
VolcanismVolcanism, vulcanism or volcanicity is the phenomenon of eruption of molten rock (magma) onto the surface of the Earth or a solid-surface planet or moon, where lava, pyroclastics, and volcanic gases erupt through a break in the surface called a vent. It includes all phenomena resulting from and causing magma within the crust or mantle of the body, to rise through the crust and form volcanic rocks on the surface. Magmas that reach the surface and solidify form extrusive landforms.
EarthEarth is the third planet from the Sun and the only astronomical object known to harbor life. This is enabled by Earth being a water world, the only one in the Solar System sustaining liquid surface water. Almost all of Earth's water is contained in its global ocean, covering 70.8% of Earth's surface. The remaining 29.2% of Earth's surface is land, most of which is located in the form of continental landmasses within one hemisphere, Earth's land hemisphere.
Layered intrusionA layered intrusion is a large sill-like body of igneous rock which exhibits vertical layering or differences in composition and texture. These intrusions can be many kilometres in area covering from around to over and several hundred metres to over in thickness. While most layered intrusions are Archean to Proterozoic in age (for example, the Paleoproterozoic Bushveld complex), they may be any age such as the Cenozoic Skaergaard intrusion of east Greenland or the Rum layered intrusion in Scotland.
Internal structure of EarthThe internal structure of Earth is the solid portion of the Earth, excluding its atmosphere and hydrosphere. The structure consists of an outer silicate solid crust, a highly viscous asthenosphere and solid mantle, a liquid outer core whose flow generates the Earth's magnetic field, and a solid inner core.
PredictionA prediction (Latin præ-, "before," and dicere, "to say"), or forecast, is a statement about a future event or data. They are often, but not always, based upon experience or knowledge. There is no universal agreement about the exact difference from "estimation"; different authors and disciplines ascribe different connotations. Future events are necessarily uncertain, so guaranteed accurate information about the future is impossible. Prediction can be useful to assist in making plans about possible developments.
