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 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 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 Hermite Type Spline Spaces over Rectangular Meshes with Complex Topological Structures

2017 ◽  
Vol 21 (3) ◽  
pp. 835-866 ◽  
Author(s):  
Meng Wu ◽  
Bernard Mourrain ◽  
André Galligo ◽  
Boniface Nkonga
Keyword(s):  
Isogeometric Analysis ◽  
Convergence Rates ◽  
Approximation Order ◽  
Basis Functions ◽  
Physical Domain ◽  
Piecewise Polynomial ◽  
Numerical Convergence ◽  
Rectangular Mesh ◽  
Piecewise Polynomial Functions ◽  
Potential Applications

AbstractMotivated by the magneto hydrodynamic (MHD) simulation for Tokamaks with Isogeometric analysis, we present splines defined over a rectangular mesh with a complex topological structure, i.e., with extraordinary vertices. These splines are piecewise polynomial functions of bi-degree (d,d) and parameter continuity. And we compute their dimension and exhibit basis functions called Hermite bases for bicubic spline spaces. We investigate their potential applications for solving partial differential equations (PDEs) over a physical domain in the framework of Isogeometric analysis. For instance, we analyze the property of approximation of these spline spaces for the L2-norm; we show that the optimal approximation order and numerical convergence rates are reached by setting a proper parameterization, although the fact that the basis functions are singular at extraordinary vertices.

Convergence behavior of popular schemes in case of calculating on adaptive grids problems with layers

2020 ◽  
pp. 66-79
Author(s):  
Владимир Дмитриевич Лисейкин ◽  
Виктор Иванович Паасонен
Keyword(s):  
Numerical Experiments ◽  
Order Equation ◽  
Second Order ◽  
Adaptive Grids ◽  
Coordinate Transformations ◽  
Diagonal Dominance ◽  
Counter Flow ◽  
Solution Quality ◽  
Uniform Grids ◽  
New Variables

Проведено сравнение качества решений модельного уравнения второго порядка с малым параметром, полученных по трем различным разностным схемам на специальных адаптивных сетках, явно задаваемых координатным преобразованием, а также на равномерных сетках в новых переменных, соответствующих этому преобразованию. Исследуются схемы второго порядка точности с диагональным преобладанием и без него и простейшая противопотоковая схема. На основе оценок погрешностей сделаны прогнозы относительно свойств решений, подтвержденные анализом и численными экспериментами. Показано, что схема второго порядка аппроксимации с диагональным преобладанием сходится равномерно по малому параметру со вторым порядком лишь в частном случае, когда коэффициент при старшей производной мал только в слое; если же он мал также и вне слоя, порядок сходимости первый. Установлено также, что схема без диагонального преобладания имеет существенно более качественные решения без осцилляций в новых переменных на равномерной сетке, чем в соответствующих им исходных физических координатах. В противоположность ей схемы с диагональным преобладанием не чувствительны к выбору системы координат. The paper compares solution quality to some model second- order equation with a small parameter obtained through three different schemes both on special adaptive grids specified explicitly by coordinate transformations eliminating layers and on uniform grids in a new coordinate related to the transformations. The schemes up to second order in physical and transformation variables both with a diagonal and not diagonal dominance and the simplest counter-flow scheme are analyzed. Predictions of a solution behavior based on estimates of solution errors are described, which are confirmed by numerical experiments and proofs. It is established, in particular, that the scheme of the second order with a diagonal dominance converges uniformly if the coefficient before the second derivative is small at the points of the boundary layer only. It was also demonstrated for the schemes without a diagonal dominance, mach better solutions without oscillations are obtained on uniform grids in new variables than on corresponding adaptive grids in the original physical coordinates.

scholarly journals Prediction of Oilfield-Increased Production Using Adaptive Neurofuzzy Inference System with Smoothing Treatment

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Lin Chen ◽  
Zhibin Liu ◽  
Nannan Ma ◽  
Yi Wang
Keyword(s):  
Initial Data ◽  
Numerical Experiments ◽  
The Novel ◽  
Inference System ◽  
Feasible Method ◽  
Prediction Result ◽  
Well Stimulation ◽  
Adaptive Neurofuzzy Inference System ◽  
Oilfield Development ◽  
Better Than

A novel modified adaptive neurofuzzy inference system with smoothing treatment (MANFIS) is proposed. The MANFIS model considered the smoothing treatment of initial data basing on the adaptive neurofuzzy inference system, and we used it to predict oilfield-increased production under the well stimulation. Numerical experiments show the prediction result of the novel considering smoothing treatment is better than that without smoothing treatment. This study provides a novel and feasible method for prediction of oilfield-increased production under well stimulation, and it can be helpful in the further study of oilfield development measure planning.

Numerical experiments on two-dimensional foam

1992 ◽  
Vol 241 ◽  
pp. 233-260 ◽  
Author(s):  
Thomas Herdtle ◽  
Hassan Aref
Keyword(s):  
Initial Data ◽  
Analytical Solutions ◽  
Numerical Experiments ◽  
Laboratory Experiments ◽  
Evolution Equations ◽  
Two Dimensional ◽  
Von Neumann ◽  
Cpu Time ◽  
Statistical Evolution ◽  
Physical Principles

The statistical evolution of a two-dimensional polygonal, or ‘dry’, foam during diffusion of gas between bubbles lends itself to a very simple mathematical description by combining physical principles discovered by Young. Laplace, Plateau, and von Neumann over a period of a century and a half. Following a brief review of this ‘canonical’ theory, we report results of the largest numerical simulations of this system undertaken to date. In particular, we discuss the existence and properties of a scaling regime, conjectured on the basis of laboratory experiments on larger systems than ours by Glazier and coworkers, and corroborated in computations on smaller systems by Weaire and collaborators. While we find qualitative agreement with these earlier investigations, our results differ on important, quantitative details, and we find that the evolution of the foam, and the emergence of scaling, is very sensitive to correlations in the initial data. The largest computations we have performed follow the relaxation of a system with 1024 bubbles to one with O(10), and took about 30 hours of CPU time on a Cray-YMP supercomputer. The code used has been thoroughly tested, both by comparison with a set of essentially analytic results on the rheology of a monodisperse-hexagonal foam due to Kraynik & Hansen, and by verification of certain analytical solutions to the evolution equations that we found for a family of ‘fractal foams’.

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 Representation of solutions to wave equations with profile functions

2020 ◽  
Vol 18 (06) ◽  
pp. 1001-1024
Author(s):  
Agnes Lamacz ◽  
Ben Schweizer
Keyword(s):  
Initial Data ◽  
Evolution Equation ◽  
Wave Equations ◽  
Dispersion Function ◽  
Fourier Space ◽  
Reconstruction Formula ◽  
Profile Equation ◽  
Representation Of Solutions ◽  
Reconstruction Operator ◽  
Constant Coefficients

In many applications, solutions to wave equations can be represented in Fourier space with the help of a dispersion function. Examples include wave equations on periodic lattices with spacing [Formula: see text], wave equations on [Formula: see text] with constant coefficients, and wave equations on [Formula: see text] with coefficients of periodicity [Formula: see text]. We characterize such solutions for large times [Formula: see text]. We establish a reconstruction formula that yields approximations for solutions in three steps: (1) From given initial data [Formula: see text], appropriate initial data for a profile equation are extracted. (2) The dispersion function determines a profile evolution equation, which, in turn, yields the shape of the profile at time [Formula: see text]. (3) A shell reconstruction operator transforms the profile to a function on [Formula: see text]. The resulting function is a good approximation of the solution [Formula: see text].

Superconvergence of a fully conservative finite difference method on non-uniform staggered grids for simulating wormhole propagation with the Darcy–Brinkman–Forchheimer framework

2019 ◽  
Vol 872 ◽  
pp. 438-471 ◽  
Author(s):  
Xiaoli Li ◽  
Hongxing Rui
Keyword(s):  
Porous Media ◽  
Finite Difference ◽  
Theoretical Analysis ◽  
Numerical Experiments ◽  
Mass Conservation ◽  
Second Order ◽  
Staggered Grids ◽  
Uniform Grids ◽  
Flux Variable ◽  
Conservative Finite Difference Method

In this paper, a finite difference scheme on non-uniform staggered grids is proposed for wormhole propagation with the Darcy–Brinkman–Forchheimer framework in porous media by introducing an auxiliary flux variable to guarantee full mass conservation. Error estimates for the pressure, velocity, porosity, concentration and auxiliary flux with second-order superconvergence in different discrete norms are established rigorously and carefully on non-uniform grids. We also obtain second-order superconvergence for some terms of the $H^{1}$ norm of the velocity on non-uniform grids. Finally, some numerical experiments are presented to verify the theoretical analysis and effectiveness of the proposed scheme.

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