An Efficient Implicit Discontinuous Spectral Galerkin Method

Rasetarinera, Patrick; Hussaini, M. Y.

doi:10.1006/jcph.2001.6853

Cited by 148 publications

(75 citation statements)

References 24 publications

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“…As in a usual FV method, the Riemann solver [46] stabilizes the solution. However in this case higher accuracy may be achieved by increasing the order of the approximation, N, as well as by reducing the size of the elements, h. The DGSEM is used in a wide range of applications such as compressible flows [5,35], electromagnetics and optics [1,13,14,29], heat transfer [32], aeroacoustics [9,36,42,43], meteorology [22,23,38], and geophysics [16,17].…”

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Quasi-A Priori Truncation Error Estimation in the DGSEM

et al. 2014

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In this paper we show how to accurately perform a quasi-a priori estimation of the truncation error of steady-state solutions computed by a discontinuous Galerkin spectral element method. We estimate the spatial truncation error using the r-estimation procedure. While most works in the literature rely on fully time-converged solutions on grids with different spacing to perform the estimation, we use non time-converged solutions on one grid with different polynomial orders. The quasi-a priori approach estimates the error while the residual of the time-iterative method is not negligible. Furthermore, the method permits one to decouple the surface and the volume contributions of the truncation error, and provides information about the anisotropy of the solution as well as its rate of convergence in polynomial order. First, we focus on the analysis of one dimensional scalar conservation laws to examine the accuracy of the estimate. Then, we extend the analysis to two dimensional problems. We demonstrate that this quasi-a priori approach yields a spectrally accurate estimate of the truncation error.

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Quasi-A Priori Truncation Error Estimation in the DGSEM

et al. 2014

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“…Several authors have applied Newton GMRES and implicit Runge-Kutta methods to the resolution of compressible Euler, Navier-Stokes and RANS equations in the context of DG discretizations [6,7,22,30,39]. Various preconditioned iterative algorithms have been introduced showing the potential benefits of taking advantage of either the block structure of the Jacobian matrix or a factorization using reordering [18,25,38,39]. In [29] a block diagonal preconditioner for linear problems based upon a domain decomposition method with static condensation is seen to achieve an optimal convergence where the convergence rate of the implicit iterative solver is independent of the number of DOF.…”

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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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“…За последние десятилетия было разработано большое количество неявных схем, основанных на методах полной и приближенной факторизации и мето-дах расщепления по физическим процессам и пространственным направлени-ям (позволяющим свести решение многомерной задачи к последовательности одномерных аналогов), в той или иной степени обеспечивающих компромисс между высокой скоростью сходимости, требованиями к памяти и эффектив-ностью параллельной реализации [2][3][4][5][6][7][8][9][10]. Настоящая работа является развитием предложенного авторами в рабо-те [11] подхода к построению эффективной параллельной реализации неяв-ной схемы на основе метода LU-SGS.…”

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Simulation of transition between regular and Mach shock waves reflections by an implicit scheme based on the LU-SGS and BiCGStab methods

Borisov

Lutsky

2016

KIAM Prepr.

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Моделирование перехода между регулярным и маховским отражением ударных волн с помощью неявной схемы на основе методов LU-SGS и BiCGStab Москва, 2016 В.Е. Борисов, А.Е. Луцкий, Моделирование перехода между регулярным и маховским отражением ударных волн с помощью неявной схемы на основе методов LU-SGS и BiCGStab Аннотация. В работе представлена параллельная реализация неявной схе-мы на основе методов LU-SGS и BiCGStab для решения трехмерных нестацио-нарных уравнений URANS с моделью турбулентности Спаларта-Аллмараса.С помощью разработанной методики проведено численное моделирование нестационарной задачи о переходе между регулярным и маховским отра-жением ударных волн, инициируемом импульсным подводом энергии в по-ток. Результаты исследований сравниваются с опубликованными в литера-туре данными, а также с результатами расчетов по явной схеме. Проведено исследование масштабируемости предложенной численной методики. 1Ключевые слова: LU-SGS, BiCGStab, URANS, вложение энергии в поток, SA модель турбулентности.V.E. Borisov, A.E. Lusky, Simulation of transition between regular and Mach shock waves reflections by an implicit scheme based on the LU-SGS and BiCGStab methods Abstract. In this paper a parallel solver for 3D unsteady Reynolds-averaged Navier-Stokes equations with Spalart-Allmaras turbulence model is presented. The method is based on fully implicit finite volume scheme, which is solved by BiCGStab with LU-SGS as precondition. Using the developed method the transition between regular and Mach shock waves reflections initiated by energy input is simulated. Numerical results are compared with published data and with results of computations by the explicit scheme. The scalability of the proposed parallel numerical method is studied.

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An Efficient Implicit Discontinuous Spectral Galerkin Method

Cited by 148 publications

References 24 publications

Quasi-A Priori Truncation Error Estimation in the DGSEM

Quasi-A Priori Truncation Error Estimation in the DGSEM

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

Simulation of transition between regular and Mach shock waves reflections by an implicit scheme based on the LU-SGS and BiCGStab methods

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