scholarly journals A Piecewise Polynomial Harmonic Nonlinear Interpolatory Reconstruction Operator on Non Uniform Grids—Adaptation around Jump Discontinuities and Elimination of Gibbs Phenomenon

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 335
Author(s):  
Pedro Ortiz ◽  
Juan Carlos Trillo
Keyword(s):  
Gibbs Phenomenon ◽  
Order Of Approximation ◽  
Piecewise Polynomial ◽  
Nonuniform Grids ◽  
Numerical Examples ◽  
Piecewise Polynomials ◽  
Jump Discontinuities ◽  
Uniform Grids ◽  
Reconstruction Operator ◽  
Numerical Order

In this paper, we analyze the behavior of a nonlinear reconstruction operator called PPH around discontinuities. The acronym PPH stands for Piecewise Polynomial Harmonic, since it uses piecewise polynomials defined by means of an adaption based on the use of the weighted Harmonic mean. This study is carried out in the general case of nonuniform grids, although for some results we restrict to σ quasi-uniform grids. In particular we analyze the numerical order of approximation close to jump discontinuities and the elimination of the Gibbs effects. We show, both theoretically and with numerical examples, that the numerical order is reduced but not completely lost as it is the case in their linear counterparts. Moreover we observe that the reconstruction is free of any Gibbs effects for sufficiently small grid sizes.

scholarly journals Analysis of a New Nonlinear Interpolatory Subdivision Scheme on σ Quasi-Uniform Grids

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1320
Author(s):  
Pedro Ortiz ◽  
Juan Carlos Trillo
Keyword(s):  
Limit Function ◽  
Subdivision Scheme ◽  
Piecewise Polynomial ◽  
Nonuniform Grids ◽  
Piecewise Polynomials ◽  
Uniform Grids ◽  
Reconstruction Operator ◽  
Nonlinear Subdivision ◽  
Subdivision Operator ◽  
Theoretical Results

In this paper, we introduce and analyze the behavior of a nonlinear subdivision operator called PPH, which comes from its associated PPH nonlinear reconstruction operator on nonuniform grids. The acronym PPH stands for Piecewise Polynomial Harmonic, since the reconstruction is built by using piecewise polynomials defined by means of an adaption based on the use of the weighted Harmonic mean. The novelty of this work lies in the generalization of the already existing PPH subdivision scheme to the nonuniform case. We define the corresponding subdivision scheme and study some important issues related to subdivision schemes such as convergence, smoothness of the limit function, and preservation of convexity. In order to obtain general results, we consider σ quasi-uniform grids. We also perform some numerical experiments to reinforce the theoretical results.

scholarly journals On the Convexity Preservation of a Quasi C3 Nonlinear Interpolatory Reconstruction Operator on σ Quasi-Uniform Grids

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 310 ◽  
Author(s):  
Pedro Ortiz ◽  
Juan Carlos Trillo
Keyword(s):  
Initial Data ◽  
Numerical Experiments ◽  
Approximation Order ◽  
Piecewise Polynomial ◽  
Nonuniform Grids ◽  
Uniform Grids ◽  
Reconstruction Operator ◽  
Convexity Properties ◽  
Second Degree

This paper is devoted to introducing a nonlinear reconstruction operator, the piecewise polynomial harmonic (PPH), on nonuniform grids. We define this operator and we study its main properties, such as its reproduction of second-degree polynomials, approximation order, and conditions for convexity preservation. In particular, for σ quasi-uniform grids with σ≤4, we get a quasi C3 reconstruction that maintains the convexity properties of the initial data. We give some numerical experiments regarding the approximation order and the convexity preservation.

scholarly journals On a class of splines free of Gibbs phenomenon

2020 ◽  
Author(s):  
Sergio Amat ◽  
Juan Ruiz ◽  
Chi-Wang Shu ◽  
Juan Carlos Trillo
Keyword(s):  
Spline Interpolation ◽  
Gibbs Phenomenon ◽  
Jump Discontinuity ◽  
Regularity Conditions ◽  
Weighted Mean ◽  
Piecewise Polynomials ◽  
Jump Discontinuities ◽  
Nonlinear Technique ◽  
The Right ◽  
Theoretical Accuracy

When interpolating data with certain regularity, spline functions are useful. They are defined as piecewise polynomials that satisfy certain regularity conditions at the joints. In the literature about splines it is possible to find several references that study the apparition of Gibbs phenomenon close to jump discontinuities in the results obtained by spline interpolation. This work is devoted to the construction and analysis of a new nonlinear technique that allows to improve the accuracy of splines near jump discontinuities eliminating the Gibbs phenomenon. The adaption is easily attained through a nonlinear modification of the right hand side of the system of equations of the spline, that contains divided differences. The modification is based on the use of a new limiter specifically designed to attain adaption close to jumps in the function. The new limiter can be seen as a nonlinear weighted mean that has better adaption properties than the linear weighted mean. We will prove that the nonlinear modification introduced in the spline keeps the maximum theoretical accuracy in all the domain except at the intervals that contain a jump discontinuity, where Gibbs oscillations are eliminated. Diffusion is introduced, but this is fine if the discontinuity appears due to a discretization of a high gradient with not enough accuracy. The new technique is introduced for cubic splines, but the theory presented allows to generalize the results very easily to splines of any order. The experiments presented satisfy the theoretical aspects analyzed in the paper.

Multivariate piecewise polynomials

Acta Numerica ◽  
1993 ◽  
Vol 2 ◽  
pp. 65-109 ◽  
Author(s):  
C. de Boor
Keyword(s):  
Approximation Theory ◽  
Variational Approach ◽  
Polynomial Function ◽  
Piecewise Polynomial ◽  
Multivariate Splines ◽  
Piecewise Polynomial Function ◽  
Multivariate Spline ◽  
Piecewise Polynomials ◽  
The Subject ◽  
Function Of Several Arguments

This article was supposed to be on ‘multivariate splines». An informal survey, taken recently by asking various people in Approximation Theory what they consider to be a ‘multivariate spline’, resulted in the answer that a multivariate spline is a possibly smooth piecewise polynomial function of several arguments. In particular the potentially very useful thin-plate spline was thought to belong more to the subject of radial basis funtions than in the present article. This is all the more surprising to me since I am convinced that the variational approach to splines will play a much greater role in multivariate spline theory than it did or should have in the univariate theory. Still, as there is more than enough material for a survey of multivariate piecewise polynomials, this article is restricted to this topic, as is indicated by the (changed) title.

Solvability of cardinal spline interpolation problems

1983 ◽  
Vol 95 (1-2) ◽  
pp. 39-57
Author(s):  
T. N. T. Goodman
Keyword(s):  
Incidence Matrix ◽  
Bounded Solution ◽  
Spline Interpolation ◽  
Piecewise Polynomial ◽  
Piecewise Polynomials ◽  
Cardinal Spline ◽  
Interpolation Problems ◽  
Bounded Data ◽  
Derivatives Of ◽  
Cardinal Spline Interpolation

SynopsisWe consider interpolation by piecewise polynomials, where the interpolation conditions are on certain derivatives of the function at certain points of a periodic vector x, specified by a periodic incidence matrix G. Similarly, we allow discontinuity of certain derivatives of the piecewise polynomial at certain points of x, specified by a periodic incidence matrix H. This generalises the well-known cardinal spline interpolation of Schoenberg. We investigate conditions on G, H and x under which there is a unique bounded solution for any given bounded data.

scholarly journals COLLOCATION APPROXIMATIONS FOR WEAKLY SINGULAR VOLTERRA INTEGRO‐DIFFERENTIAL EQUATIONS

2003 ◽  
Vol 8 (4) ◽  
pp. 315-328 ◽  
Author(s):  
I. Parts ◽  
A. Pedas
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Order Of Convergence ◽  
Piecewise Polynomial ◽  
Volterra Type ◽  
Weakly Singular ◽  
Uniform Grids ◽  
Attainable Order ◽  
Piecewise Polynomial Collocation Method ◽  
Polynomial Collocation

A piecewise polynomial collocation method for solving linear weakly singular integro‐differential equations of Volterra type is constructed. The attainable order of convergence of collocation approximations on arbitrary and quasi‐uniform grids is studied theoretically and numerically.

scholarly journals Gibbs phenomena for Lq-best approximation in finite element spaces

2021 ◽  
Author(s):  
Sarah Roggendorf ◽  
Paul Houston ◽  
Kristoffer van der Zee
Keyword(s):  
Finite Element ◽  
Sufficient Conditions ◽  
Gibbs Phenomenon ◽  
Two Dimensions ◽  
Thin Layers ◽  
Jump Discontinuities ◽  
Recent Developments ◽  
Minimum Residual ◽  
Gibbs Phenomena ◽  
Theoretical Results

Recent developments in the context of minimum residual finite element methods are paving the way for designing quasi-optimal discretisation methods in non-standard function spaces, such as L q -type Sobolev spaces. For q → 1, these methods have demonstrated huge potential in avoiding the notorious Gibbs phenomena, i.e., the occurrence of spurious non-physical oscillations near thin layers and jump discontinuities. In this work we provide theoretical results that explain some of the numerical observations. In particular, we investigate the Gibbs phenomena for L q -best approximations of discontinuities in finite element spaces with 1 ≤ q < ∞. We prove sufficient conditions on meshes in one and two dimensions such that over- and undershoots vanish in the limit q → 1. Moreover, we include examples of meshes such that Gibbs phenomena remain present even for q = 1 and demonstrate that our results can be used to design meshes so as to eliminate the Gibbs phenomenon.

Multiscale Simulations of Heat Transfer and Fluid Flow Problems

2012 ◽  
Vol 134 (3) ◽  
Author(s):  
Ya-Ling He ◽  
Wen-Quan Tao
Keyword(s):  
Multiscale Simulation ◽  
Multiscale Simulations ◽  
Research Needs ◽  
Mesoscale Simulation ◽  
Numerical Examples ◽  
Multiscale Problems ◽  
Flow Problems ◽  
Reconstruction Operator ◽  
Entire Flow ◽  
Numerical Approaches

The multiscale problems in the thermal and fluid science are classified into two categories: multiscale process and multiscale system. The meanings of the two categories are described. Examples are provided for multiscale process and multiscale system. In this paper, focus is put on the simulation of multiscale process. The numerical approaches for multiscale processes have two categories: one is the usage of a general governing equation and solving the entire flow field involving a variation of several orders in characteristic geometric scale. The other is the so-called “solving regionally and coupling at the interfaces.” In this approach, the processes at different length levels are simulated by different numerical methods and then information is exchanged at the interfaces between different regions. The key point is the establishment of the reconstruction operator, which transforms the data of few variables of macroscopic computation to a large amount of variables of microscale or mesoscale simulation. Six numerical examples of multiscale simulation are presented. Finally, some research needs are proposed.

Effect of Nonuniform Grids on High-Order Finite Difference Method

2017 ◽  
Vol 9 (4) ◽  
pp. 1012-1034 ◽  
Author(s):  
Dan Xu ◽  
Xiaogang Deng ◽  
Yaming Chen ◽  
Guangxue Wang ◽  
Yidao Dong
Keyword(s):  
Finite Difference ◽  
Coordinate Transformation ◽  
High Order ◽  
Numerical Schemes ◽  
Nonuniform Grids ◽  
Curvilinear Grids ◽  
Uniform Grids ◽  
The One ◽  
High Order Finite Difference ◽  
Stretching Ratio

AbstractThe finite difference (FD) method is popular in the computational fluid dynamics and widely used in various flow simulations. Most of the FD schemes are developed on the uniform Cartesian grids; however, the use of nonuniform or curvilinear grids is inevitable for adapting to the complex configurations and the coordinate transformation is usually adopted. Therefore the question that whether the characteristics of the numerical schemes evaluated on the uniform grids can be preserved on the nonuniform grids arises, which is seldom discussed. Based on the one-dimensional wave equation, this paper systematically studies the characteristics of the high-order FD schemes on nonuniform grids, including the order of accuracy, resolution characteristics and the numerical stability. Especially, the Fourier analysis involving the metrics is presented for the first time and the relation between the resolution of numerical schemes and the stretching ratio of grids is discussed. Analysis shows that for smooth varying grids, these characteristics can be generally preserved after the coordinate transformation. Numerical tests also validate our conclusions.

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