scholarly journals Discontinuous Galerkin Discretization with Embedded Boundary Conditions

2003 ◽  
Vol 3 (1) ◽  
pp. 135-158 ◽  
Author(s):  
P. W. Hemker ◽  
W. Hoffman ◽  
M. H. Van Raalte
Keyword(s):  
Boundary Conditions ◽  
Discontinuous Galerkin ◽  
Complex Structure ◽  
High Order ◽  
Dirichlet Boundary Conditions ◽  
High Order Accuracy ◽  
Elliptic Pdes ◽  
Curved Boundary ◽  
Cubic Polynomials ◽  
Convection Dominated

AbstractThe purpose of this paper is to introduce discretization methods of discontinuous Galerkin type for solving second-order elliptic PDEs on a structured, regular rectangular grid, while the problem is defined on a curved boundary. The methods aim at high-order accuracy and the difficulty arises since the regular grid cannot follow the curved boundary. Starting with the Lagrange multiplier formulation for the boundary conditions, we derive variational forms for the discretization of 2-D elliptic problems with embedded Dirichlet boundary conditions. Within the framework of structured, regular rectangular grids, we treat curved boundaries according to the principles that underlie the discontinuous Galerkin method. Thus, the high-order DGdiscretization is adapted in cells with embedded boundaries. We give examples of approximation with tensor products of cubic polynomials. As an illustration, we solve a convection-dominated boundary-value problem on a complex domain. Although, of course, it is impossible to accurately represent a boundary layer with a complex structure by means of cubic polynomials, the boundary condition treatment appears quite effective in handling such complex situations.

scholarly journals On Source Terms and Boundary Conditions Using Arbitrary High Order Discontinuous Galerkin Schemes

2007 ◽  
Vol 17 (3) ◽  
pp. 297-310 ◽  
Author(s):  
Michael Dumbser ◽  
Claus-Dieter Munz
Keyword(s):  
Boundary Conditions ◽  
Discontinuous Galerkin ◽  
High Order ◽  
Source Terms ◽  
Riemann Problems ◽  
Time Step ◽  
Flux Function ◽  
Numerical Flux ◽  
Generalized Riemann Problems ◽  
Very High

On Source Terms and Boundary Conditions Using Arbitrary High Order Discontinuous Galerkin SchemesThis article is devoted to the discretization of source terms and boundary conditions using discontinuous Galerkin schemes with an arbitrary high order of accuracy in space and time for the solution of hyperbolic conservation laws on unstructured triangular meshes. The building block of the method is a particular numerical flux function at the element interfaces based on the solution of Generalized Riemann Problems (GRPs) with piecewise polynomial initial data. The solution of the generalized Riemann problem, originally introduced by Toro and Titarev in a finite volume context, provides simultaneously a numerical flux function as well as a time integration method. The resulting scheme is extremely local since it integrates the PDE from one time step to the successive one in a single step using only information from the direct side neighbors. Since source terms are directly incorporated into the numerical flux via the solution of the GRP, our very high order accurate method is also able to maintain very well smooth steady-state solutions of PDEs with source terms, similar to the so-called well-balanced schemes which are usually specially designed for this purpose. Boundary conditions are imposed solving inverse generalized Riemann problems. Furthermore, we show numerical evidence proving that by using very high order schemes together with high order polynomial representations of curved boundaries, high quality solutions can be obtained on very coarse meshes.

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.

scholarly journals A Sub-element Adaptive Shock Capturing Approach for Discontinuous Galerkin Methods

2021 ◽  
Author(s):  
Johannes Markert ◽  
Gregor Gassner ◽  
Stefanie Walch
Keyword(s):  
Discontinuous Galerkin ◽  
Adaptive Mesh Refinement ◽  
Adaptive Mesh ◽  
High Order ◽  
Shock Capturing ◽  
High Order Accuracy ◽  
Finite Volume Scheme ◽  
Galerkin Methods ◽  
Order Accuracy ◽  
New Strategy

AbstractIn this paper, a new strategy for a sub-element-based shock capturing for discontinuous Galerkin (DG) approximations is presented. The idea is to interpret a DG element as a collection of data and construct a hierarchy of low-to-high-order discretizations on this set of data, including a first-order finite volume scheme up to the full-order DG scheme. The different DG discretizations are then blended according to sub-element troubled cell indicators, resulting in a final discretization that adaptively blends from low to high order within a single DG element. The goal is to retain as much high-order accuracy as possible, even in simulations with very strong shocks, as, e.g., presented in the Sedov test. The framework retains the locality of the standard DG scheme and is hence well suited for a combination with adaptive mesh refinement and parallel computing. The numerical tests demonstrate the sub-element adaptive behavior of the new shock capturing approach and its high accuracy.

Novel Adaptive Hybrid Discontinuous Galerkin Algorithms for Elliptic Problems

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Dongwook Shin ◽  
Yoongu Hwang ◽  
Eun-Jae Park
Keyword(s):  
Discontinuous Galerkin ◽  
Elliptic Problems ◽  
Posteriori Error ◽  
Optimal Choice ◽  
High Order ◽  
High Order Accuracy ◽  
Error Estimator ◽  
New Strategy ◽  
Data Oscillation ◽  
Second Order Elliptic Problems

Abstract In this work, we develop novel adaptive hybrid discontinuous Galerkin algorithms for second-order elliptic problems. For this, two types of reliable and efficient, modulo a data-oscillation term, and fully computable a posteriori error estimators are developed: the first one is a simple residual type error estimator, and the second is a flux reconstruction based error estimator of a guaranteed type for polynomial approximations of any degree by using a simple postprocessing. These estimators can achieve high-order accuracy for both smooth and nonsmooth problems even with high-order approximations. In order to enhance the performance of adaptive algorithms, we introduce 𝐾-means clustering based marking strategy. The choice of marking parameter is crucial in the performance of the existing strategy such as maximum and bulk criteria; however, the optimal choice is not known. The new strategy has no unknown parameter. Several numerical examples are given to illustrate the performance of the new marking strategy along with our estimators.

scholarly journals Implementation and Verification of High-Order Accurate Discontinuous Galerkin Method on 2-D Unstructured Grids

2013 ◽  
Vol 432 ◽  
pp. 221-226
Author(s):  
Wen Geng Zhao ◽  
Hong Wei Zheng
Keyword(s):  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Experimental Tests ◽  
High Order ◽  
High Order Accuracy ◽  
Order Accuracy ◽  
2D Euler Equation ◽  
Unstructured Girds ◽  
Complex Boundaries

In this paper, the discontinuous Galerkin method (DG) is applied to solve the 2D Euler equation. DG can be easily used in the unstructured girds, which has advantages in dealing with problems with complex boundaries. High order accuracy is achieved by higher order polynomial approximations within elements. In order to capture the shock without oscillation, the limiter is also applied. The performance of DG is illustrated by three numerical experimental tests, which show the potential of DG in engineering applications. The vortex propagation problem is to verify high-order accuracy of DG, while Sob problem and forward step problem are used to illustrate the ability to capture shock.

High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (II)

2015 ◽  
Vol 18 (3) ◽  
Author(s):  
Jianxiong Cao ◽  
Changpin Li ◽  
YangQuan Chen
Keyword(s):  
Dirichlet Boundary ◽  
Order Approximation ◽  
High Order ◽  
Dirichlet Boundary Conditions ◽  
High Order Accuracy ◽  
High Order Approximation ◽  
Order Accuracy ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
New Formula

AbstractIn this paper, we first establish a high-order numerical algorithm for α-th (0 < α < 1) order Caputo derivative of a given function f(t), where the convergence rate is (4 − α)-th order. Then by using this new formula, an improved difference scheme with high order accuracy in time to solve Caputo-type fractional advection-diffusion equation with Dirichlet boundary conditions is constructed. Finally, numerical examples are carried out to confirm the efficiency of the constructed algorithm.

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