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Compact and Stable Discontinuous Galerkin Methods for Convection-Diffusion Problems

S. Brdar, A. Dedner, R. Klöfkorn
2012 SIAM Journal on Scientific Computing  
We present a new scheme, the compact discontinuous Galerkin 2 (CDG2) method, for solving nonlinear convection-diffusion problems together with a detailed comparison to other well-accepted DG methods.  ...  More recently the compact discontinuous Galerkin (CDG) method was introduced in [36].  ...  The authors would like to thank the reviewers and the editor for their various important remarks and suggestions.  ... 
doi:10.1137/100817528 fatcat:t2b5bv6yn5g2fekxiadojmqvue

A Conservative DGM for Convection-Diffusion and Navier-Stokes Problems [chapter]

J. Tinsley Oden, Carlos Erik Baumann
2000 Lecture Notes in Computational Science and Engineering  
An hp-adaptive con~ervative Discontinuous Galerkin Method for the solution of convection-diffusion problems is reviewed.  ...  The variational formulation for diffusion terms produces a compact, locally conservative, higher-order accurate, and stable solver.  ...  Discontinuous Galerkin approximation The discontinuous Galerkin formulation for convection-diffusion problems is built as an extension of the classical discontinuous Galerkin method for hyperbolic problems  ... 
doi:10.1007/978-3-642-59721-3_13 fatcat:42by7ntfxvgvpm2o5ihwrqelcq

Page 7153 of Mathematical Reviews Vol. , Issue 97K [page]

1997 Mathematical Reviews  
Appropriate numerical treatment of convective and diffusive terms becomes essential especially when dealing with convection-dominated diffusion problems.  ...  Alternatively, the convection-diffusion equations can be discretized by using semi-implicit methods. An implicitness parameter @ is introduced for each term of the convection-diffusion equation.  ... 

Page 8469 of Mathematical Reviews Vol. , Issue 2002K [page]

2002 Mathematical Reviews  
Runge-Kutta discontinuous Galerkin methods.  ...  ; for moving shocks the discontinuous Galerkin method gives a better resolution.  ... 

Page 8938 of Mathematical Reviews Vol. , Issue 2000m [page]

2000 Mathematical Reviews  
Methods Engrg. 47 (2000), no. 1-3, 61-73. Summary: “We present an adaptive-order discontinuous Galerkin technique that produces a compact, higher-order-accurate, and stable solver.  ...  C. (4-WALS-NM; Swansea) Implicit characteristic Galerkin method for convection-diffusion equations. (English summary) Internat. J. Numer. Methods Engrg. 47 (2000), no. 10, 1689-1708.  ... 

Page 5082 of Mathematical Reviews Vol. , Issue 2000g [page]

2000 Mathematical Reviews  
The authors solve a time-dependent convection-diffusion problem using a spectral Lagrange-Galerkin method.  ...  F. (3-CALG-MS; Calgary, AB) Stability and convergence of the spectral Lagrange-Galerkin method for mixed periodic/non-periodic convection-dominated diffusion problems. (English summary) IMA J. Numer.  ... 

Page 2760 of Mathematical Reviews Vol. , Issue 2002D [page]

2002 Mathematical Reviews  
Summary: “An /p-adaptive conservative discontinuous Galerkin method for the solution of convection-diffusion problems is re- viewed.  ...  Summary: “The discontinuous Galerkin method (DGM) and con- tinuous Galerkin method (CGM) are investigated and compared for the advection problem and the diffusion problem.  ... 

Page 7047 of Mathematical Reviews Vol. , Issue 99j [page]

1999 Mathematical Reviews  
The authors deal with the application of the local discontinuous Galerkin method to unsteady, nonlinear, convection-diffusion sys- tem of equations.  ...  Qian Li (PRC-SHDN; Jinan) 99j:65163 65M60 76M25 Cockburn, Bernardo (1-MN-SM; Minneapolis, MN); Shu, Chi-Wang (1-BRN-A; Providence, RI) The local discontinuous Galerkin method for time-dependent convection-diffusion  ... 


Jue Yan, Chi-Wang Shu
2012 Journal of Scientific Computing  
We review local discontinuous Galerkin methods for convection diffusion equations involving second derivatives and for KdV type equations involving third derivatives.  ...  In this paper we review the existing and develop new local discontinuous Galerkin methods for solving time dependent partial differential equations with higher order derivatives in one and multiple space  ...  The first local discontinuous Galerkin method was developed by Cockburn and Shu [11] , for the convection diffusion equation (1.2) containing second derivatives.  ... 
doi:10.1023/a:1015132126817 fatcat:ep2u55csdrgjvdtxf7djlvuvaq

Page 3744 of Mathematical Reviews Vol. , Issue 2003e [page]

2003 Mathematical Reviews  
In this paper, a local pointwise error estimate is presented for the lumped characteristics-Galerkin method for the one-dimensional linear convection-dominated diffusion problem.  ...  convection-diffusion problems.  ... 

High order WENO and DG methods for time-dependent convection-dominated PDEs: A brief survey of several recent developments

Chi-Wang Shu
2016 Journal of Computational Physics  
Galerkin (DG) finite element methods, emphasizing several of their recent developments: bound-preserving limiters for DG, finite volume and finite difference schemes, which address issues in robustness  ...  In this article we give a brief survey of two selected classes of high order methods, namely the weighted essentially non-oscillatory (WENO) finite difference and finite volume schemes and discontinuous  ...  Baumann and J.T. Oden, A discontinuous hp finite element method for convection-diffusion problems, Computer Methods in Applied Mechanics and Engineering, 175 (1999), 311-341. [10] G. Bechon, R.  ... 
doi:10.1016/ fatcat:ucrmechmcfcitdjhvb3ynd2g6e

Page 4165 of Mathematical Reviews Vol. , Issue 2001F [page]

2001 Mathematical Reviews  
{For the entire collection see MR 2001d:65004. } 2001f:65117 65M70 65-02 76M22 76N10 Cockburn, Bernardo (1-MN-SM; Minneapolis, MN) An introduction to the discontinuous Galerkin method for convection-dominated  ...  The scheme is then extended to multiple space dimensions and to the case of convection-diffusion systems.  ... 

A discontinuoushp finite element method for the Euler and Navier-Stokes equations

Carlos Erik Baumann, J. Tinsley Oden
1999 International Journal for Numerical Methods in Fluids  
This paper introduces a new method for the solution of the Euler and Navier-Stokes equations, which is based on the application of a recently developed discontinuous Galerkin technique to obtain a compact  ...  , higher-order accurate and stable solver.  ...  Note that n represents the outward normal of the element with higher index_ THE DISCONTINUOUS GALERKIN METHOD The discontinuous Galerkin formulation for convection-diffusion problems is built as an extension  ... 
doi:10.1002/(sici)1097-0363(19990915)31:1<79::aid-fld956>;2-c fatcat:cz5l27csyfc25ox7ozopv3dkiu

Local Discontinuous Galerkin Methods for the Degasperis-Procesi Equation

Yan Xu, Chi-Wang Shu
2011 Communications in Computational Physics  
AbstractIn this paper, we develop, analyze and test local discontinuous Galerkin (LDG) methods for solving the Degasperis-Procesi equation which contains nonlinear high order derivatives, and possibly  ...  The LDG method has the flexibility for arbitrary h and p adaptivity. We prove the L2stability for general solutions.  ...  Discontinuous Galerkin methods for convection-dominated problems. In T. Barth and H.  ... 
doi:10.4208/cicp.2009.09.023 fatcat:jwudzqy6w5e3jcm6bpjjy4shey

Automatic Variationally Stable Analysis for FE Computations: An Introduction [article]

Victor M. Calo, Albert Romkes, Eirik Valseth
2019 arXiv   pre-print
The results are not highly diffused for convection-dominated problems ...  ...  We introduce an automatic variationally stable analysis (AVS) for finite element (FE) computations of scalar-valued convection-diffusion equations with non-constant and highly oscillatory coefficients.  ...  In this manuscript, we introduce the automatic variationally stable (AVS) analysis for FE computations of the convection-diffusion equation in which the diffusion and convection coefficients can be highly  ... 
arXiv:1808.01888v3 fatcat:3lsfjxdpyfflxkvxj4csem5roi
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