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Strang Splitting
Strang splitting is a numerical method for solving differential equations that are decomposable into a sum of differential operators. It is named after Gilbert Strang. It is used to speed up calculation for problems involving operators on very different time scales, for example, chemical reactions in fluid dynamics, and to solve multidimensional partial differential equations by reducing them to a sum of one-dimensional problems. Fractional step methods As a precursor to Strang splitting, consider a differential equation of the form : \frac = L_1 () + L_2 () where L_1, L_2 are differential operators. If L_1 and L_2 were constant coefficient matrices, then the exact solution to the associated initial value problem would be : y(t) = e^ y_0. If L_1 and L_2 commute, then by the exponential laws this is equivalent to : y(t) = e^ e^ y_0. If they do not, then by the Baker–Campbell–Hausdorff formula it is still possible to replace the exponential of the sum by a product of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly. Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gilbert Strang
William Gilbert Strang (born November 27, 1934), usually known as simply Gilbert Strang or Gil Strang, is an American mathematician, with contributions to finite element theory, the calculus of variations, wavelet analysis and linear algebra. He has made many contributions to mathematics education, including publishing seven mathematics textbooks and one monograph. Strang is the MathWorks Professor of Mathematics at the Massachusetts Institute of Technology. He teaches Introduction to Linear Algebra, Computational Science and Engineering, and Matrix Methods, and his lectures are freely available through MIT OpenCourseWare. Education Strang completed his undergraduate degree ( S.B.) in 1955 from Massachusetts Institute of Technology. He was a member of the Theta Deuteron Charge of Theta Delta Chi fraternity. He was the recipient of Rhodes Scholarship from University of Oxford, where he received his B.A. and M.A. from Balliol College in 1957. Strang earned his Ph. D. from U ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to Numerical methods for partial differential equations, numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematics, pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function (in the style of a higher-order function in computer science). This article considers mainly linear differential operators, which are the most common type. However, non-linear differential operators also exist, such as the Schwarzian derivative. Definition An order-m linear differential operator is a map A from a function space \mathcal_1 to another function space \mathcal_2 that can be written as: A = \sum_a_\alpha(x) D^\alpha\ , where \alpha = (\alpha_1,\alpha_2,\cdots,\alpha_n) is a multi-index of non-negative integers, , \alpha, = \alpha_1 + \alpha_2 + \cdots + \alpha_n, and for each \alpha, a_\alpha(x) is a function on some open domain in ''n''-dimensional space. The operator D^\alpha is interpreted as D^\alp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Operator Splitting Topics
This is a list of operator splitting topics. General *Alternating direction implicit method — finite difference method for parabolic, hyperbolic, and elliptic partial differential equations *GRADELA — simple gradient elasticity model *Matrix splitting — general method of splitting a matrix operator into a sum or difference of matrices *Paul Tseng — resolved question on convergence of matrix splitting algorithms *PISO algorithm — pressure-velocity calculation for Navier-Stokes equations *Projection method (fluid dynamics) — computational fluid dynamics method *Reactive transport modeling in porous media — modeling of chemical reactions and fluid flow through the Earth's crust *Richard S. Varga — developed matrix splitting *Strang splitting Strang splitting is a numerical method for solving differential equations that are decomposable into a sum of differential operators. It is named after Gilbert Strang. It is used to speed up calculation for problems involving opera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Splitting
In the mathematical discipline of numerical linear algebra, a matrix splitting is an expression which represents a given matrix as a sum or difference of matrices. Many iterative methods (for example, for systems of differential equations) depend upon the direct solution of matrix equations involving matrices more general than tridiagonal matrices. These matrix equations can often be solved directly and efficiently when written as a matrix splitting. The technique was devised by Richard S. Varga in 1960. Regular splittings We seek to solve the matrix equation where A is a given ''n'' × ''n'' non-singular matrix, and k is a given column vector with ''n'' components. We split the matrix A into where B and C are ''n'' × ''n'' matrices. If, for an arbitrary ''n'' × ''n'' matrix M, M has nonnegative entries, we write M ≥ 0. If M has only positive entries, we write M > 0. Similarly, if the matrix M1 − M2 has nonnegative entries, we write M1 ≥ M2. Definit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert McLachlan (mathematician)
Robert Iain McLachlan (born 1964) is a New Zealand mathematician and Distinguished Professor in the School of Fundamental Sciences, Massey University, New Zealand. His research in geometric integration encompasses both pure and applied mathematics, modelling the structure of systems such as liquids, climate cycles, and quantum mechanics. He is also writes for the public on the subject of climate change policy. Academic career McLachlan was born in Christchurch, New Zealand in 1964, and studied mathematics at the University of Canterbury, graduating with a BSc (Hons) First Class in 1984. One formative experience was in his last year of high school, where he had free rein to experiment with assembly language programming on the school PDP-11/10. McLachlan went on to graduate work in numerical analysis in 1986. He received a PhD from Caltech (the California Institute of Technology) in 1990, supervised by Herbert Keller in computational fluid dynamics, with a thesis titled "Sepa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Randall J
Randall may refer to the following: Places United States *Randall, California, former name of White Hall, California, an unincorporated community * Randall, Indiana, a former town *Randall, Iowa, a city *Randall, Kansas, a city *Randall, Minnesota, a city * Randall, West Virginia, an unincorporated community *Randall, Wisconsin, a town *Randall, Burnett County, Wisconsin, an unincorporated community *Randall County, Texas * Randall Creek, in Nebraska and South Dakota *Randall's Island, part of New York City *Camp Randall, Madison, Wisconsin, a former army camp, on the National Register of Historic Places *Fort Randall, South Dakota, a former military base, on the National Register of Historic Places Elsewhere *Mount Randall, Victoria Land, Antarctica *Randall Rocks, Graham Land, Antarctica *Randall, a community in the town of New Tecumseth, Ontario, Canada Businesses *Randall Amplifiers, a manufacturer of guitar amplifiers *Randall House Publications, American publisher *Randall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
