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

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.

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

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.

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

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.

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

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.

Representation of solutions to wave equations with profile functions

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].

A Hybrid High-Order Method for Highly Oscillatory Elliptic Problems

We devise a Hybrid High-Order (HHO) method for highly oscillatory elliptic problems that is capable of handling general meshes. The method hinges on discrete unknowns that are polynomials attached to the faces and cells of a coarse mesh; those attached to the cells can be eliminated locally using static condensation. The main building ingredient is a reconstruction operator, local to each coarse cell, that maps onto a fine-scale space spanned by oscillatory basis functions. The present HHO method generalizes the ideas of some existing multiscale approaches, while providing the first complete analysis on general meshes. It also improves on those methods, taking advantage of the flexibility granted by the HHO framework. The method handles arbitrary orders of approximation{k\geq 0}. For face unknowns that are polynomials of degreek, we devise two versions of the method, depending on the polynomial degree{(k-1)}orkof the cell unknowns. We prove, in the case of periodic coefficients, an energy-error estimate of the form{(\varepsilon^{\frac{1}{2}}+H^{k+1}+(\frac{\varepsilon}{H})^{\frac{1}{2}})}, and we illustrate our theoretical findings on some test-cases.

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

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.