scholarly journals Stable Filtering Procedures for Nodal Discontinuous Galerkin Methods

2021 ◽  
Vol 87 (1) ◽  
Author(s):  
Jan Nordström ◽  
Andrew R. Winters
Keyword(s):  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Methods ◽  
Finite Difference Methods ◽  
Basis Functions ◽  
Galerkin Methods ◽  
Theoretical Discussion ◽  
Numerical Tests ◽  
Dg Methods ◽  
Theoretical Results ◽  
Filtering Procedure

AbstractWe prove that the most common filtering procedure for nodal discontinuous Galerkin (DG) methods is stable. The proof exploits that the DG approximation is constructed from polynomial basis functions and that integrals are approximated with high-order accurate Legendre–Gauss–Lobatto quadrature. The theoretical discussion re-contextualizes stable filtering results for finite difference methods into the DG setting. Numerical tests verify and validate the underlying theoretical results.

High-Order Mesh Generation for Discontinuous Galerkin Methods Based on Elastic Deformation

2016 ◽  
Vol 8 (4) ◽  
pp. 693-702
Author(s):  
Hongqiang Lu ◽  
Kai Cao ◽  
Lechao Bian ◽  
Yizhao Wu
Keyword(s):  
Finite Element Method ◽  
Discontinuous Galerkin ◽  
Mesh Generation ◽  
Discontinuous Galerkin Methods ◽  
Elasticity Modulus ◽  
High Order ◽  
Galerkin Methods ◽  
Governing Equations ◽  
Numerical Tests ◽  
Cubic Polynomial

AbstractIn this paper, a high-order curved mesh generation method for Discontinuous Galerkin methods is introduced. First, a regular mesh is generated. Second, the solid surface is re-constructed using cubic polynomial. Third, the elastic governing equations are solved using high-order finite element method to provide a fully or partly curved grid. Numerical tests indicate that the intersection between element boundaries can be avoided by carefully defining the elasticity modulus.

scholarly journals Layer-Wise Discontinuous Galerkin Methods for Piezoelectric Laminates

Modelling ◽  
2020 ◽  
Vol 1 (2) ◽  
pp. 198-214
Author(s):  
Ivano Benedetti ◽  
Vincenzo Gulizzi ◽  
Alberto Milazzo
Keyword(s):  
Discontinuous Galerkin ◽  
Electric Potential ◽  
Discontinuous Galerkin Methods ◽  
Basis Functions ◽  
Variable Order ◽  
Test Cases ◽  
Galerkin Methods ◽  
Computational Tool ◽  
Interior Penalty ◽  
Analysis And Design

In this work, a novel high-order formulation for multilayered piezoelectric plates based on the combination of variable-order interior penalty discontinuous Galerkin methods and general layer-wise plate theories is presented, implemented and tested. The key feature of the formulation is the possibility to tune the order of the basis functions in both the in-plane approximation and the through-the-thickness expansion of the primary variables, namely displacements and electric potential. The results obtained from the application to the considered test cases show accuracy and robustness, thus confirming the developed technique as a supplementary computational tool for the analysis and design of smart laminated devices.

Discrete maximum principle for interior penalty discontinuous Galerkin methods

2013 ◽  
Vol 11 (4) ◽  
Author(s):  
Tamás Horváth ◽  
Miklós Mincsovics
Keyword(s):  
Maximum Principle ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Methods ◽  
Mesh Size ◽  
Discrete Maximum Principle ◽  
Galerkin Methods ◽  
Penalty Parameter ◽  
Discrete Level ◽  
Interior Penalty ◽  
Theoretical Results

AbstractA class of linear elliptic operators has an important qualitative property, the so-called maximum principle. In this paper we investigate how this property can be preserved on the discrete level when an interior penalty discontinuous Galerkin method is applied for the discretization of a 1D elliptic operator. We give mesh conditions for the symmetric and for the incomplete method that establish some connection between the mesh size and the penalty parameter. We then investigate the sharpness of these conditions. The theoretical results are illustrated with numerical examples.

A Straightforward hp-Adaptivity Strategy for Shock-Capturing with High-Order Discontinuous Galerkin Methods

2014 ◽  
Vol 6 (01) ◽  
pp. 135-144 ◽  
Author(s):  
Hongqiang Lu ◽  
Qiang Sun
Keyword(s):  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Methods ◽  
Mesh Size ◽  
High Order ◽  
Shock Capturing ◽  
Local Order ◽  
Galerkin Methods ◽  
Numerical Tests ◽  
Dg Method ◽  
Coarse Grids

AbstractIn this paper, high-order Discontinuous Galerkin (DG) method is used to solve the two-dimensional Euler equations. A shock-capturing method based on the artificial viscosity technique is employed to handle physical discontinuities. Numerical tests show that the shocks can be captured within one element even on very coarse grids. The thickness of the shocks is dominated by the local mesh size and the local order of the basis functions. In order to obtain better shock resolution, a straightforwardhp-adaptivity strategy is introduced, which is based on the high-order contribution calculated using hierarchical basis. Numerical results indicate that thehp-adaptivity method is easy to implement and better shock resolution can be obtained with smaller local mesh size and higher local order.

STABILIZED DISCONTINUOUS GALERKIN METHODS FOR FLOW–SOUND INTERACTION

2007 ◽  
Vol 15 (01) ◽  
pp. 123-143 ◽  
Author(s):  
ANDREAS RICHTER ◽  
JÖRG STILLER ◽  
ROGER GRUNDMANN
Keyword(s):  
Wave Propagation ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Methods ◽  
Musical Instruments ◽  
Galerkin Methods ◽  
Numerical Tests ◽  
Characteristic Boundary ◽  
Boundary Treatment ◽  
Classical Characteristic ◽  
Local Extrapolation

The wave propagation in musical instruments is considered. A high-order discontinuous Galerkin method for solving the Euler equations is presented, and an alternative concept for supplying local non-reflecting boundary conditions is introduced. This concept is based on local extrapolation in conjunction with a stabilizing slope limiting. The practicability of this method is discussed via numerical tests, and the results are compared with classical characteristic boundary treatment. Investigations of wave propagation phenomena in musical instruments like the bassoon are performed, and the results are presented.

Optimal error estimates of the semidiscrete discontinuous Galerkin methods for two dimensional hyperbolic equations on Cartesian meshes using Pk elements

2020 ◽  
Vol 54 (2) ◽  
pp. 705-726
Author(s):  
Yong Liu ◽  
Chi-Wang Shu ◽  
Mengping Zhang
Keyword(s):  
Error Estimates ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Methods ◽  
Hyperbolic Equations ◽  
Optimal Error Estimates ◽  
Galerkin Methods ◽  
Optimal Error ◽  
Piecewise Polynomials ◽  
Cartesian Meshes ◽  
Theoretical Results

In this paper, we study the optimal error estimates of the classical discontinuous Galerkin method for time-dependent 2-D hyperbolic equations using Pk elements on uniform Cartesian meshes, and prove that the error in the L2 norm achieves optimal (k + 1)th order convergence when upwind fluxes are used. For the linear constant coefficient case, the results hold true for arbitrary piecewise polynomials of degree k ≥ 0. For variable coefficient and nonlinear cases, we give the proof for piecewise polynomials of degree k = 0, 1, 2, 3 and k = 2, 3, respectively, under the condition that the wind direction does not change. The theoretical results are verified by numerical examples.

scholarly journals A Novel Robust Strategy for Discontinuous Galerkin Methods in Computational Fluid Mechanics: Why? When? What? Where?

2021 ◽  
Vol 8 ◽  
Author(s):  
Gregor J. Gassner ◽  
Andrew R. Winters
Keyword(s):  
Wave Propagation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Methods ◽  
Computational Physics ◽  
High Order ◽  
Galerkin Methods ◽  
Partial Differential ◽  
Dg Methods

In this paper we will review a recent emerging paradigm shift in the construction and analysis of high order Discontinuous Galerkin methods applied to approximate solutions of hyperbolic or mixed hyperbolic-parabolic partial differential equations (PDEs) in computational physics. There is a long history using DG methods to approximate the solution of partial differential equations in computational physics with successful applications in linear wave propagation, like those governed by Maxwell’s equations, incompressible and compressible fluid and plasma dynamics governed by the Navier-Stokes and the Magnetohydrodynamics equations, or as a solver for ordinary differential equations (ODEs), e.g., in structural mechanics. The DG method amalgamates ideas from several existing methods such as the Finite Element Galerkin method (FEM) and the Finite Volume method (FVM) and is specifically applied to problems with advection dominated properties, such as fast moving fluids or wave propagation. In the numerics community, DG methods are infamous for being computationally complex and, due to their high order nature, as having issues with robustness, i.e., these methods are sometimes prone to crashing easily. In this article we will focus on efficient nodal versions of the DG scheme and present recent ideas to restore its robustness, its connections to and influence by other sectors of the numerical community, such as the finite difference community, and further discuss this young, but rapidly developing research topic by highlighting the main contributions and a closing discussion about possible next lines of research.

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