Analysis of the discontinuous Galerkin method for nonlinear convection–diffusion problems

Dolejší, Vít; Feistauer, Miloslav; Sobotíková, Veronika

doi:10.1016/j.cma.2004.07.017

Cited by 63 publications

(58 citation statements)

References 10 publications

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“…In recent years, there has been a strong interest for these techniques in the field of computational fluid dynamics which has led to the introduction of discretization schemes for parabolic and purely elliptic equations. See for example [4,5,8,9,13,19,20,24,28,37,43] and references cited therein. For more details, the reader is referred to the analysis of existing discretizations in an unified framework developed by Arnold et al [3] and to the overview of recent progress in DG methods for compressible flows [32].…”

mentioning

confidence: 99%

Time Implicit High-Order Discontinuous Galerkin Method with Reduced Evaluation Cost

Renac¹,

Marmignon²,

Coquel³

2012

SIAM J. Sci. Comput.

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To cite this version:Florent Renac, Claude Marmignon, Frédéric Coquel.Time implicit high-order discontinuous galerkin method with reduced evaluation cost.SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2012, 34 (1) Abstract. An efficient and robust time integration procedure for a high-order discontinuous Galerkin method is introduced for solving nonlinear second-order partial differential equations. The time discretization is based on an explicit formulation for the hyperbolic term and an implicit formulation for the parabolic term. The procedure uses an iterative algorithm with reduced evaluation cost. The size of the linear system to be solved is greatly reduced thanks to partial uncoupling in space between low-order and high-order degrees of freedom. Numerical examples are presented for the nonlinear convection-diffusion equation in one and two dimensions including steady and unsteady flow problems. The performance of the present method is investigated in terms of CPU time and compared to a fully implicit method. A Von Neumann stability analysis is carried out in order to determine the stability and damping properties of the method. Besides a fairly reduced CPU effort, numerical results demonstrate better convergence properties of the present algorithm when compared to the fully implicit method.Key words. discontinuous Galerkin method, nonlinear convection-diffusion equation, implicitexplicit time discretization, pseudo-time integration AMS subject classifications. 65N30, 65N121. Introduction. Discontinuous Galerkin (DG) methods are high-order finite element discretizations and were introduced in the early 1970s for the numerical simulation of the first-order hyperbolic neutron transport equation [34,40]. The method was later extended to nonlinear convection dominated flow problems with the use of a Runge-Kutta method for the time integration [12,14]. In recent years, there has been a strong interest for these techniques in the field of computational fluid dynamics which has led to the introduction of discretization schemes for parabolic and purely elliptic equations. See for example [4,5,8,9,13,19,20,24,28,37,43] and references cited therein. For more details, the reader is referred to the analysis of existing discretizations in an unified framework developed by Arnold et al. [3] and to the overview of recent progress in DG methods for compressible flows [32]. The success of these methods lies in their high-order of accuracy and flexibility thanks to their high degree of locality. The stencil of most DG methods is compact and independent of the space discretization order. This means that the evaluation of the residual in a discretization element involves only this element and its immediate neighbours. The order of the numerical scheme depends on the degree of the approximated piecewise polynomials which can be easily increased. These properties make the DG method well suited to algorithm parallelization, hp-refinement, hp-multigrid, unstructured meshes, the applicati...

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confidence: 99%

Time Implicit High-Order Discontinuous Galerkin Method with Reduced Evaluation Cost

Renac¹,

Marmignon²,

Coquel³

2012

SIAM J. Sci. Comput.

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“…Comparing this estimate with results from [1] we see that our estimate is better due to the linearity of our problem.…”

Section: Theorem Let Us Assume That {T H } H∈(0h 0 ) Is a System Of mentioning

confidence: 56%

Discontinuous Galerkin method of lines for solving nonstationary singularly perturbed linear problems

Feistauer

Svadlenka

2004

Journal of Numerical Mathematics

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“…• analysis of optimal error estimates for problems with nonlinear convection and diffusion, • analysis of error estimates of the DGFEM applied to problems with mixed boundary conditions on nonstandard meshes, used in [10] and [8].…”

Section: Discussionmentioning

confidence: 99%

“…Similarly to [7] or [10] we can show that a sufficiently regular exact solution u of problem (2.1) satisfies the identity…”

Section: Discontinuous Galerkin Space Semidiscretizationmentioning

confidence: 99%

“…In our recent papers [7], [10], [9] we applied the IPG methods to a scalar nonstationary convection-diffusion equation with nonlinear Lipschitz continuous convective terms. This equation represents a model problem for the solution of the system of the compressible Navier-Stokes equations which describes the flow of viscous compressible fluids.…”

Section: Introductionmentioning

confidence: 99%

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Discontinuous Galerkin method for nonlinear convection-diffusion problems with mixed Dirichlet-Neumann boundary conditions

2010

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Analysis of the discontinuous Galerkin method for nonlinear convection–diffusion problems

Cited by 63 publications

References 10 publications

Time Implicit High-Order Discontinuous Galerkin Method with Reduced Evaluation Cost

Time Implicit High-Order Discontinuous Galerkin Method with Reduced Evaluation Cost

Discontinuous Galerkin method of lines for solving nonstationary singularly perturbed linear problems

Discontinuous Galerkin method for nonlinear convection-diffusion problems with mixed Dirichlet-Neumann boundary conditions

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