scholarly journals A Space-Time Interior Penalty Discontinuous Galerkin Method for the Wave Equation

2022 ◽  
Author(s):  
Poorvi Shukla ◽  
J. J. W. van der Vegt
Keyword(s):  
Wave Equation ◽  
Discontinuous Galerkin ◽  
Mesh Size ◽  
Space Time ◽  
Basis Functions ◽  
Optimal Order ◽  
Time Step ◽  
Order Of Accuracy ◽  
Interior Penalty ◽  
Dg Method

AbstractA new higher-order accurate space-time discontinuous Galerkin (DG) method using the interior penalty flux and discontinuous basis functions, both in space and in time, is presented and fully analyzed for the second-order scalar wave equation. Special attention is given to the definition of the numerical fluxes since they are crucial for the stability and accuracy of the space-time DG method. The theoretical analysis shows that the DG discretization is stable and converges in a DG-norm on general unstructured and locally refined meshes, including local refinement in time. The space-time interior penalty DG discretization does not have a CFL-type restriction for stability. Optimal order of accuracy is obtained in the DG-norm if the mesh size h and the time step $$\Delta t$$ Δ t satisfy $$h\cong C\Delta t$$ h ≅ C Δ t , with C a positive constant. The optimal order of accuracy of the space-time DG discretization in the DG-norm is confirmed by calculations on several model problems. These calculations also show that for pth-order tensor product basis functions the convergence rate in the $$L^\infty$$ L ∞ and $$L^2$$ L 2 -norms is order $$p+1$$ p + 1 for polynomial orders $$p=1$$ p = 1 and $$p=3$$ p = 3 and order p for polynomial order $$p=2$$ p = 2 .

A Compact High Order Space-Time Method for Conservation Laws

2011 ◽  
Vol 9 (2) ◽  
pp. 441-480 ◽  
Author(s):  
Shuangzhang Tu ◽  
Gordon W. Skelton ◽  
Qing Pang
Keyword(s):  
Conservation Laws ◽  
Discontinuous Galerkin ◽  
Hyperbolic Conservation Laws ◽  
Space Time ◽  
High Order ◽  
Basis Functions ◽  
High Order Accuracy ◽  
Time Step ◽  
Order Of Accuracy ◽  
Hyperbolic Conservation

AbstractThis paper presents a novel high-order space-time method for hyperbolic conservation laws. Two important concepts, the staggered space-time mesh of the space-time conservation element/solution element (CE/SE) method and the local discontinuous basis functions of the space-time discontinuous Galerkin (DG) finite element method, are the two key ingredients of the new scheme. The staggered space-time mesh is constructed using the cell-vertex structure of the underlying spatial mesh. The universal definitions of CEs and SEs are independent of the underlying spatial mesh and thus suitable for arbitrarily unstructured meshes. The solution within each physical time step is updated alternately at the cell level and the vertex level. For this solution updating strategy and the DG ingredient, the new scheme here is termed as the discontinuous Galerkin cell-vertex scheme (DG-CVS). The high order of accuracy is achieved by employing high-order Taylor polynomials as the basis functions inside each SE. The present DG-CVS exhibits many advantageous features such as Riemann-solver-free, high-order accuracy, point-implicitness, compactness, and ease of handling boundary conditions. Several numerical tests including the scalar advection equations and compressible Euler equations will demonstrate the performance of the new method.

COMPUTATIONAL ASPECTS OF THE DISCONTINUOUS GALERKIN METHOD FOR THE WAVE EQUATION

2008 ◽  
Vol 16 (04) ◽  
pp. 507-530 ◽  
Author(s):  
TIMO LÄHIVAARA ◽  
MATTI MALINEN ◽  
JARI P. KAIPIO ◽  
TOMI HUTTUNEN
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Wave Propagation ◽  
Wave Equation ◽  
Discontinuous Galerkin ◽  
Homogeneous Medium ◽  
Polynomial Degree ◽  
Dg Method ◽  
Runge Kutta ◽  
Element Method

The Discontinuous Galerkin (DG) method is a powerful tool for numerically simulating wave propagation problems. In this paper, the time-dependent wave equation is solved using the DG method for spatial discretization; and the Crank–Nicolson and fourth-order explicit, singly diagonally implicit Runge–Kutta methods, and, for reference, the explicit Runge–Kutta method, were used for time integration. These simulation methods were studied using two-dimensional numerical experiments. The aim of the experiments was to study the effect of the polynomial degree of the basis functions, grid density, and the Courant–Friedrichs–Lewy number on the accuracy of the approximation. The sensitivity of the methods to distorted finite elements was also examined. Results from the DG method were compared with those computed using a conventional finite element method. Three different model problems were considered. In the first experiment, wave propagation in a homogeneous medium was studied. In the second experiment, the scattering and propagation of waves in an inhomogeneous medium were investigated. The third experiment evaluated wave propagation in a more complicated domain involving multiple scattering waves. The results indicated that the DG method provides more accurate solutions than the conventional finite element method with a reduced computation time and a lower number of degrees of freedom.

scholarly journals Analysis of discontinuous Galerkin dual-primal isogeometric tearing and interconnecting methods

2017 ◽  
Vol 28 (01) ◽  
pp. 131-158 ◽  
Author(s):  
Christoph Hofer
Keyword(s):  
Discontinuous Galerkin ◽  
Condition Number ◽  
Mesh Size ◽  
Penalty Term ◽  
Constant Diffusion Coefficient ◽  
The Poisson Equation ◽  
Constant Diffusion ◽  
Dg Method ◽  
Optimal Bounds ◽  
Two Space Dimensions

In this paper, we present the analysis of the discontinuous Galerkin dual-primal isogeometric tearing and interconnecting method (dG-IETI-DP) for a multipatch discretization in two-space dimensions where we only consider vertex primal variables. As model problem, we use the Poisson equation with globally constant diffusion coefficient. The dG-IETI-DP method is a combination of the dual-primal isogeometric tearing and interconnecting method (IETI-DP) with the discontinuous Galerkin (dG) method. We use the dG method only on the interfaces to couple different patches. This enables us to handle non-matching grids on patch interfaces as well as segmentation crimes (gaps and overlaps) between the patches. The purpose of this paper is to derive quasi-optimal bounds for the condition number of the preconditioned system with respect to the maximal ratio [Formula: see text] of subdomain diameter and mesh size. Moreover, we show that the condition number is independent of the number of patches, but depends on the mesh sizes of neighboring patches [Formula: see text] and the parameter [Formula: see text] in the dG penalty term.

Local Space-Time Adaptive Discontinuous Galerkin Finite Element Methods for Time-Dependent Waves

2003 ◽  
Author(s):  
Lonny L. Thompson ◽  
Dantong He
Keyword(s):  
Discontinuous Galerkin ◽  
Acoustic Radiation ◽  
Adaptive Strategy ◽  
Adaptive Methods ◽  
Space Time ◽  
Time Dependent ◽  
Time Interval ◽  
Time Step ◽  
Continuous Space ◽  
Local Space

Local space-time adaptive methods are developed including high-order accurate nonreflecting boundary conditions (NRBC) for time-dependent waves. The time-discontinuous Galerkin (TDG) variational method is used to divide the time-interval into space-time slabs, the solution advanced from one slab to the next. Within each slab, a continuous space-time mesh is used which enables local sub-time steps. By maintaining orthogonality of the space-time mesh and pre-integrating analytically through the time-slab, we obtain an efficient yet robust local space-time adaptive method. Any standard spatial element may be used together with standard spatial mesh generation and visualization methods. Recovery based error estimates are used in both space and time dimensions to determine the number and size of local space-time elements within a global time step such that both the spatial and temporal estimated error is equally distributed throughout the space-time approximation. The result is an efficient and reliable adaptive strategy which distributes local space-time elements where needed to accurately track time-dependent waves over large distances and time. Numerical examples of time-dependent acoustic radiation are given which demonstrate the accuracy, reliability and efficiency gained from this new technology.

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.

An Iterative Discontinuous Galerkin Method for Solving the Nonlinear Poisson Boltzmann Equation

2014 ◽  
Vol 16 (2) ◽  
pp. 491-515 ◽  
Author(s):  
Peimeng Yin ◽  
Yunqing Huang ◽  
Hailiang Liu
Keyword(s):  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Solution Space ◽  
Initial Guess ◽  
Two Dimensional ◽  
Order Of Accuracy ◽  
Dg Method ◽  
Poisson Boltzmann ◽  
Poisson Boltzmann Equation

AbstractAn iterative discontinuous Galerkin (DG) method is proposed to solve the nonlinear Poisson Boltzmann (PB) equation. We first identify a function space in which the solution of the nonlinear PB equation is iteratively approximated through a series of linear PB equations, while an appropriate initial guess and a suitable iterative parameter are selected so that the solutions of linear PB equations are monotone within the identified solution space. For the spatial discretization we apply the direct discontinuous Galerkin method to those linear PB equations. More precisely, we use one initial guess when the Debye parameterλ=(1), and a special initial guess forλ≫1 to ensure convergence. The iterative parameter is carefully chosen to guarantee the existence, uniqueness, and convergence of the iteration. In particular, iteration steps can be reduced for a variable iterative parameter. Both one and two-dimensional numerical results are carried out to demonstrate both accuracy and capacity of the iterative DG method for both cases ofλ=(1) andλ≪ 1. The (m+ 1)th order of accuracy forL2andmth order of accuracy forH1forPmelements are numerically obtained.

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