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.

Fitted mesh methods for problems with parabolic boundary layers

Abstract: A Dirichlet boundary value problem for a linear parabolic dierential equation is studied on a rectangular domain in the x t plane. The coecient of the second order space derivative is a small singular perturbation parameter, which gives rise to parabolic boundary layers on the two lateral sides of the rectangle. It is proved that a numerical method, comprising a standard nite dierence operator (centred in space, implicit in time) on a tted piecewise uniform mesh of NxNt elements condensing in the boundary layers, is uniform with respect to the small parameter, in the sense that its numerical solutions converge in the maximum norm to the exact solution uniformly well for all values of the parameter in the semi-open interval (0,1]. More specically, it is shown that the errors are bounded in the maximum norm by C((N 1 x lnNx) 2 +N 1 t ), where C is a constant independent not only of Nx and Nt but also of the small parameter. Numerical results are presented, which validate numerically this theoretical result and show that a numerical method consisting of the same nite dierence operator on a uniform mesh of NxNt elements is not uniform with respect to the small parameter.

GENSPECT: a line-by-line code with selectable interpolation error tolerance

Abstract: Current line-by-line radiative transfer codes accelerate calculations by interpolating the line function where it varies slowly. This can increase calculation performance by a factor of 10 or more but causes a reduction in calculation accuracy. We present a new line-by-line algorithm that computes absorption coefficients to a specified percentage-error tolerance in a near minimal number of calculations. The algorithm employs a novel binary division of a calculation's spectral interval, coupled with a pre-computed lookup table that predicts where it is appropriate to reduce the resolution of a particular line without exceeding the required error tolerance. Line contributions are computed piecewise across a cascaded series of grids which are then interpolated and summed to derive the absorption coefficient. The algorithm is coded in MATLAB as part of a toolbox of radiative transfer functions for the analysis of planetary atmospheres and laboratory experiments.

Refinement and coarsening indicators for adaptive parametrization: application to the estimation of hydraulic transmissivities

Abstract: When estimating hydraulic transmissivity the question of parametrization is of great importance. The transmissivity is assumed to be a piecewise constant space-dependent function and the unknowns are both the transmissivity values and the zonation, the partition of the domain whose parts correspond to the zones where the transmissivity is constant. Refinement and coarsening indicators, which are easy to compute from the gradient of the least squares misfit function, are introduced to construct iteratively the zonation and to prevent overparametrization.

On the Fourier transform of the indicator function of a planar set.

Abstract: Suppose C is a compact subset of the plane having a piecewise smooth boundary AC. Let F(r, 0) be the Fourier transform, in polar coordinates, of the indicator function of the set C, where by the indicator function of C, we mean the function whose value on C is 1, and whose value on the complement of C is 0. In ?1 of this paper, we shall describe some relationships between geometric properties of C, and the asymptotic behavior of F(r, 0) as r -x- 00. In ?2, we shall give applications of the results of ?1 to some questions in the geometry of numbers. 1. If AC is sufficiently smooth, and has everywhere positive Gaussian curvature, it is known that the function 'D(6) = sup, r312IF(r, 0)1 is bounded on S' (cf. [1]). If AC has points of zero curvature, this need no longer be true (cf. [3]). The following, however, remains true: THEOREM 1. If AC is of class Cn + 3, for some integer n 1, and if the Gaussian curvature of AC is nonzero at all points of AC, with the possible exception of a finite set, at each point of which the tangent line has contact of order 1. Moreover, 'D(6) is always bounded, except in neighborhoods of those points of S' which, regarded as vectors, correspond to exterior or interior normals to AC at points of zero curvature. In a neighborhood of such a point 006 'D(6) is bounded by a multiple of [dist (6, 00)] -n - 1)/2n , where dist (6, 00) is the length of the smaller arc of S' connecting 0 and 6o, and nj is the largest order of contact which can occur between AC and its tangent line, at those points of AC at which the exterior normal is either 00 or- o REMARK. Theorem 1 has analogues in higher dimensions. I shall show in another paper, by different methods, that if C is a compact convex subset of Rn, whose boundary is analytic, and if F(r, 0) is the Fourier transform, in polar coordinates, of the indicator function of the set C, then supr r(n + 1)/21 F(r, 0)1 is of class LP on Sn1, for some p>2. If C is a polygon, the estimates are of a quite different character. THEOREM 2. Suppose C is a polygon. Then