scholarly journals On certain inequalities associated to curvature properties of the nonlinear PPH reconstruction operator

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
J. C. Trillo ◽  
P. Ortiz
Keyword(s):  
Reconstruction Operator

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 Collocation Methods for High-Order Well-Balanced Methods for Systems of Balance Laws

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1799
Author(s):  
Irene Gómez-Bueno ◽  
Manuel Jesús Castro Díaz ◽  
Carlos Parés ◽  
Giovanni Russo
Keyword(s):  
High Order ◽  
Collocation Methods ◽  
Quadrature Formulas ◽  
Balance Laws ◽  
Source Terms ◽  
Time Step ◽  
Systems Of Balance Laws ◽  
Reconstruction Operator ◽  
Non Linear ◽  
Linear Problems

In some previous works, two of the authors introduced a technique to design high-order numerical methods for one-dimensional balance laws that preserve all their stationary solutions. The basis of these methods is a well-balanced reconstruction operator. Moreover, they introduced a procedure to modify any standard reconstruction operator, like MUSCL, ENO, CWENO, etc., in order to be well-balanced. This strategy involves a non-linear problem at every cell at every time step that consists in finding the stationary solution whose average is the given cell value. In a recent paper, a fully well-balanced method is presented where the non-linear problems to be solved in the reconstruction procedure are interpreted as control problems. The goal of this paper is to introduce a new technique to solve these local non-linear problems based on the application of the collocation RK methods. Special care is put to analyze the effects of computing the averages and the source terms using quadrature formulas. A general technique which allows us to deal with resonant problems is also introduced. To check the efficiency of the methods and their well-balance property, they have been applied to a number of tests, ranging from easy academic systems of balance laws consisting of Burgers equation with some non-linear source terms to the shallow water equations—without and with Manning friction—or Euler equations of gas dynamics with gravity effects.

A CONSTRUCTION OF SPACES OF COMPATIBLE DIFFERENTIAL FORMS ON CELLULAR COMPLEXES

2008 ◽  
Vol 18 (05) ◽  
pp. 739-757 ◽  
Author(s):  
SNORRE H. CHRISTIANSEN
Keyword(s):  
Finite Element ◽  
Mixed Finite Element ◽  
Differential Forms ◽  
Cochain Complex ◽  
Two Dimensional ◽  
Exterior Derivative ◽  
Whitney Forms ◽  
Reconstruction Operator ◽  
Mimetic Finite Differences ◽  
Cellular Complex

Given a cellular complex, we construct spaces of differential forms which form a complex under the exterior derivative, which is isomorphic to the cochain complex of the cellular complex. The construction applies in particular to subsets of Euclidean space divided into polyhedra, for which it provides, for each k, a space of k-forms with a basis indexed by the set of k-dimensional cells. In the framework of mimetic finite differences, the construction provides a conforming reconstruction operator. The construction requires auxiliary spaces of differential forms on each cell, for which we provide two examples. When the cells are simplexes, the construction can be used to recover the standard mixed finite element spaces also called Whitney forms. We can also recover the dual finite elements previously constructed by A. Buffa and the author on the barycentric refinement of a two-dimensional mesh.

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].

scholarly journals Hybrid Discontinuous Galerkin methods with relaxed H(div)-conformity for incompressible flows. Part II

2019 ◽  
Vol 53 (2) ◽  
pp. 503-522 ◽  
Author(s):  
Philip L. Lederer ◽  
Christoph Lehrenfeld ◽  
Joachim Schöberl
Keyword(s):  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Methods ◽  
Incompressible Flows ◽  
Stokes Equations ◽  
Stokes Problem ◽  
Integration Scheme ◽  
Galerkin Methods ◽  
Robust Analysis ◽  
Reconstruction Operator ◽  
The Impact

The present work is the second part of a pair of papers, considering Hybrid Discontinuous Galerkin methods with relaxed H(div)-conformity. The first part mainly dealt with presenting a robust analysis with respect to the mesh size h and the introduction of a reconstruction operator to restore divergence-conformity and pressure robustness (pressure independent velocity error estimates) using a modified force discretization. The aim of this part is the presentation of a high order polynomial robust analysis for the relaxed H(div)-conforming Hybrid Discontinuous Galerkin discretization of the two dimensional Stokes problem. It is based on the recently proven polynomial robust LBB-condition for BDM elements, Lederer and Schöberl (IMA J. Numer. Anal. (2017)) and is derived by a direct approach instead of using a best approximation Céa like result. We further treat the impact of the reconstruction operator on the hp analysis and present a numerical investigation considering polynomial robustness. We conclude the paper presenting an efficient operator splitting time integration scheme for the Navier–Stokes equations which is based on the methods recently presented in Lehrenfeld and Schöberl (Comp. Methods Appl. Mech. Eng. 307 (2016) 339–361) and includes the ideas of the reconstruction operator.

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.

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