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Piecewise

About: Piecewise is a research topic. Over the lifetime, 21064 publications have been published within this topic receiving 432096 citations. The topic is also known as: piecewise-defined function & hybrid function.


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01 Dec 1998
TL;DR: In this article, a solution-mapping procedure is applied to parametrize the solution of the initial-value ordinary differential equation system as a set of algebraic polynomial equations.
Abstract: In a chemical kinetics calculation, a solution-mapping procedure is applied to parametrize the solution of the initial-value ordinary differential equation system as a set of algebraic polynomial equations. To increase the accuracy, the parametrization is done piecewise, dividing the multidimensional chemical composition space into hypercubes and constructing polynomials for each hypercube. A differential equation solver is used to provide the solution at selected points throughout a hypercube, and from these solutions the polynomial coefficients are determined. Factorial design methods are used to reduce the required number of computed points. The polynomial coefficients for each hypercube are stored in a data structure for subsequent reuse, since over the duration of a flame simulation it is likely that a particular set of concentrations and temperature will occur repeatedly at different times and positions. The method is applied to H2–air combustion using an 8-species reaction set. After N2 is added as an inert species and enthalpy is considered, this results in a 10-dimensional chemical composition space. To add the capability of using a variable time-step, time-step is added as an additional dimension, making an 11-dimensional space. Reactive fluid dynamical simulations of a 1-D laminar premixed flame and a 2-D turbulent non-premixed jet are performed. The results are compared to identical control runs which use an ordinary differential equation solver to calculate the chemical kinetic rate equations. The resulting accuracy is very good, and a factor of 10 increase in computational efficiency is attained.

102 citations

Journal ArticleDOI
TL;DR: In this article, the consequences of discontinuities on the specification property of interval maps are studied and a necessary and sufficient condition for a piecewise monotonic, piecewise continuous map to have this property is given, and it is shown that for a large and natural class of families of such maps (including the,3-transformations), the set of parameters for which the property holds, though dense, has zero Lebesgue measure.
Abstract: We study the consequences of discontinuities on the specification property for interval maps. After giving a necessary and sufficient condition for a piecewise monotonic, piecewise continuous map to have this property, we show that for a large and natural class of families of such maps (including the ,3-transformations), the set of parameters for which the specification property holds, though dense, has zero Lebesgue measure. Thus, regarding the specification property, the general case is at the opposite of the continuous case solved by A.M. Blokh (Russian Math. Surveys 38 (1983), 133-134) (for which we give a proof).

102 citations

Posted Content
TL;DR: In this paper, a discontinuous Galerkin method for approximating the Vlasov-Poisson system of equations describing the time evolution of a collisionless plasma is proposed, which preserves the positivity of the electron distribution function and weakly enforces continuity of the electric field through mesh interfaces and boundary conditions.
Abstract: A discontinuous Galerkin method for approximating the Vlasov-Poisson system of equations describing the time evolution of a collisionless plasma is proposed. The method is mass conservative and, in the case that piecewise constant functions are used as a basis, the method preserves the positivity of the electron distribution function and weakly enforces continuity of the electric field through mesh interfaces and boundary conditions. The performance of the method is investigated by computing several examples and error estimates associated system's approximation are stated. In particular, computed results are benchmarked against established theoretical results for linear advection and the phenomenon of linear Landau damping for both the Maxwell and Lorentz distributions. Moreover, two nonlinear problems are considered: nonlinear Landau damping and a version of the two-stream instability are computed. For the latter, fine scale details of the resulting long-time BGK-like state are presented. Conservation laws are examined and various comparisons to theory are made. The results obtained demonstrate that the discontinuous Galerkin method is a viable option for integrating the Vlasov-Poisson system.

101 citations

Journal ArticleDOI
TL;DR: In this paper, an algebraic-geometric method for nonlinear integrable PDE's is shown to lead to new piecewise smooth weak solutions of a class of N-component systems of nonlinear evolution equations.
Abstract: An extension of the algebraic-geometric method for nonlinear integrable PDE's is shown to lead to new piecewise smooth weak solutions of a class of N-component systems of nonlinear evolution equations. This class includes, among others, equations from the Dym and shallow water equation hierarchies. The main goal of the paper is to give explicit theta-functional expressions for piecewise smooth weak solutions of these nonlinear PDE's, which are associated to nonlinear subvarieties of hyperelliptic Jacobians. The main results of the present paper are twofold. First, we exhibit some of the special features of integrable PDE's that admit piecewise smooth weak solutions, which make them different from equations whose solutions are globally meromorphic, such as the KdV equation. Second, we blend the techniques of algebraic geometry and weak solutions of PDE's to gain further insight into, and explicit formulas for, piecewise-smooth finite-gap solutions. The basic technique used to achieve these aims is rather different from earlier papers dealing with peaked solutions. First, profiles of the finite-gap piecewise smooth solutions are linked to certain finite dimensional billiard dynamical systems and ellipsoidal billiards. Second, after reducing the solution of certain finite dimensional Hamiltonian systems on Riemann surfaces to the solution of a nonstandard Jacobi inversion problem, this is resolved by introducing new parametrizations. Amongst other natural consequences of the algebraic-geometric approach, we find finite dimensional integrable Hamiltonian dynamical systems describing the motion of peaks in the finite-gap as well as the limiting (soliton) cases, and solve them exactly. The dynamics of the peaks is also obtained by using Jacobi inversion problems. Finally, we relate our method to the shock wave approach for weak solutions of wave equations by determining jump conditions at the peak location.

101 citations

Journal ArticleDOI
TL;DR: Analytical methods for truncating and scaling lattices to be used in vector quantizations are given, and their utility is demonstrated for independent and identically distributed Gaussian and Laplacian sources.
Abstract: Lattice vector quantizer design procedures for nonuniform sources are presented. The procedures yield lattice vector quantizers with excellent performance and retaining the structure required for fast quantization. Analytical methods for truncating and scaling lattices to be used in vector quantizations are given, and their utility is demonstrated for independent and identically distributed (i.i.d.) Gaussian and Laplacian sources. An analytical technique for piecewise linear multidimensional compandor designs is evaluated for i.i.d. Gaussian and Laplacian sources by comparing its performance to that of the other vector quantizers. >

101 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20251
2023917
20222,014
20211,089
20201,147
20191,106