A two-stage fourth-order discontinuous Galerkin method based on the GRP solver for the compressible euler equations

Cheng, Jian; Du, Zhifang; Lei, Xin; Wang, Yue; Li, Jiequan

doi:10.1016/j.compfluid.2019.01.025

Cited by 16 publications

(8 citation statements)

References 35 publications

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“…The efficiency is mainly attributed to the half of reconstruction steps compared to that for the same order of other line methods. This is further verified in the framework of DG methods [14]. In Table 2 and Figure 6 through simulating shock-vortex interaction problem, demonstrating that nearly 55% CPU time can be saved using the GRP-DG(s2p3) method compared to the same order SSP RKDG(s5p3) method.…”

Section: Computational Efficiencymentioning

confidence: 54%

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Two-stage fourth order: temporal-spatial coupling in computational fluid dynamics (CFD)

2019

Adv. Aerodyn.

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With increasing engineering demands, there need high order accurate schemes embedded with precise physical information in order to capture delicate small scale structures and strong waves with correct"physics". There are two families of high order methods: One is the method of line, relying on the Runge-Kutta (R-K) time-stepping. The building block is the Riemann solution labeled as the solution element "1". Each step in R-K just has first order accuracy. In order to derive a fourth order accuracy scheme in time, one needs four stages labeled as "1 ⊙ 1 ⊙ 1 ⊙ 1 = 4". The other is the one-stage Lax-Wendroff (LW) type method, which is more compact but is complicated to design numerical fluxes and hard to use when applied to highly nonlinear problems. In recent years, the pair of solution element and dynamics element, labeled as "2", are taken as the building block. The direct adoption of the dynamics implies the inherent temporal-spatial coupling. With this type of building blocks, a family of two-stage fourth order accurate schemes, labeled as "2⊙2 = 4", are designed for the computation of compressible fluid flows. The resulting schemes are compact, robust and efficient. This paper contributes to elucidate how and why high order accurate schemes should be so designed. To some extent, the "2 ⊙ 2 = 4" algorithm extracts the advantages of the method of line and one-stage LW method. As a core part, the pair "2" is expounded and LW solver is revisited. The generalized Riemann problem (GRP) solver, as the discontinuous and nonlinear version of LW flow solver, and the gas kinetic scheme (GKS) solver, the microscopic LW solver, are all reviewed. The compact Hermite-type data reconstruction and high order approximation of 2 Two-stage fourth order: Temporal Spatial Coupling in CFD boundary conditions are proposed. Besides, the computational performance and prospective discussions are presented.Keywords Compressible fluid dynamics · hyperbolic balance laws · high order methods · temporal-spatial coupling · multi-stage two-derivative methods · Lax-Wendroff type flow solvers · GRP solver

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Section: Computational Efficiencymentioning

confidence: 54%

“…When applied to hyperbolic problems (16) and (22), one can formulate them in any appropriate framework such as finite volume framework [32] or discontinuous Galerkin (DG) framework [14]. Hence we assume that the computational domain Ω is meshed as Ω = ∪ j∈J Ω j and formulate the problem in the form…”

Section: Stage 1 Define Intermediate Valuesmentioning

confidence: 99%

Two-stage fourth order: temporal-spatial coupling in computational fluid dynamics (CFD)

2019

Adv. Aerodyn.

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“…Theoretically, a close coupling between the spatial and temporal evolution is recovered through the analysis of detailed wave interactions in the GRP scheme. The GRP method has been applied successfully to develop high resolution schemes and used for many engineering problems, see, e.g., [5,13,44] and the references therein.…”

Section: Grp Finite Volume Methodsmentioning

confidence: 99%

“…Finally, in Section 5, we present numerical simulations obtained by two finite volume methods: a recently proposed finite volume method, see [25], that is (entropy)-stable and consistent and thus yields the consistent approximate solutions required by the abstract theory and a more standard finite volume method based on the generalized Riemann problem, see, e.g. [4][5][6][7]13] and the references therein. The predicted strong convergence of the Cesàro averages when approximate solutions experience oscillations is confirmed by both numerical methods.…”

Section: Introductionmentioning

confidence: 99%

On oscillatory solutions to the complete Euler system

Feireisl

Klingenberg

Kreml

et al. 2020

Journal of Differential Equations

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We develop a method to compute effectively the Young measures associated to sequences of numerical solutions of the compressible Euler system. Our approach is based on the concept of K−convergence adapted to sequences of parametrized measures. The convergence is strong in space and time (a.e. pointwise or in certain L q spaces) whereas the measures converge narrowly or in the Wasserstein distance to the corresponding limit.

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“…Fortunately, the two-stage fourth-order time-stepping method has been developed for Lax-Wendroff type flow solvers [38,39,40] by using the flux and its first-order time derivative which seems simpler. Recently, the two-stage temporal discretization has also been extended to DG based on the flux solver for the generalized Riemann problem (GRP) for the Euler equations [41]. As only one middle stage is used and little additional computational cost for the time derivative is required, the scheme shows higher efficiency than the traditional RKDG.…”

Section: Introductionmentioning

confidence: 99%

A third-order gas-kinetic CPR method for the Euler and Navier–Stokes equations on triangular meshes

Zhang

et al. 2018

Journal of Computational Physics

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A two-stage fourth-order discontinuous Galerkin method based on the GRP solver for the compressible euler equations

Cited by 16 publications

References 35 publications

Two-stage fourth order: temporal-spatial coupling in computational fluid dynamics (CFD)

Two-stage fourth order: temporal-spatial coupling in computational fluid dynamics (CFD)

On oscillatory solutions to the complete Euler system

A third-order gas-kinetic CPR method for the Euler and Navier–Stokes equations on triangular meshes

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