A compact fourth-order gas-kinetic scheme for the Euler and Navier–Stokes equations

Ji, Xing; Pan, Liang; Shyy, Wei; Xu, Kun

doi:10.1016/j.jcp.2018.06.034

Cited by 56 publications

(57 citation statements)

References 62 publications

(90 reference statements)

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“…In the past years, the gas-kinetic scheme (GKS) has been developed systematically [54,53]. The gas evolution model used in this paper for the flux and interface flow variable evaluation is almost identical to that of the fourth-order compact scheme [21]. Therefore, only a brief introduction about GKS will be presented in this section.…”

Section: Gas-kinetic Scheme and Multi-stage Multi-derivative Methodsmentioning

confidence: 99%

“…With the above time accurate gas distribution function at a cell interface, in order to improve the temporal accuracy of the scheme, the multi-stages multi-derivative (MSMD) technique will be used for the evaluation of the flux function and the interface values [21]. The implementation of MSMD needs the flux function and its time derivative.…”

Section: Multi-stage Multi-derivative Time Stepping Methodsmentioning

confidence: 99%

“…Due to the use of only two reconstructions in the 4thorder scheme, the GKS becomes very efficient in comparison with schemes based on the Runge-Kutta time stepping technique. By combining the second-order or third-order GKS fluxes and MSMD technique again, a family of higher-order gas-kinetic methods has been constructed [22,21].…”

Section: Introductionmentioning

confidence: 99%

“…The time dependent gas-distribution function in GKS at the cell interface provides not only the flux and its time derivative, but also time accurate flow variables at the cell in-3 terface. The design of compact GKS based on the cell averaged and cell interface values has been conducted [52,37,38,21]. In this paper, we are going to develop 6th-order and 8th-order compact GKS in rectangular mesh for compressible flow simulations.…”

Section: Introductionmentioning

confidence: 99%

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Compact higher-order gas-kinetic schemes with spectral-like resolution for compressible flow simulations

Zhao

Shyy

et al. 2019

Adv. Aerodyn.

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In this paper, a class of compact higher-order gas-kinetic schemes (GKS) with spectral resolution will be presented. Based on the high-order gas evolution model in GKS, both the interface flux function and conservative flow variables can be evaluated explicitly from the time-accurate gas distribution function. As a result, inside each control volume both the cell-averaged flow variables and their cell-averaged gradients can be updated within each time step. The flow variable update and slope update are coming from the same physical solution at the cell interface. Different from many other approaches, such as HWENO, there are no additional governing equations in GKS for the slopes or equivalent degrees of freedom independently inside each cell. Therefore, based on both cell averaged values and their slopes, compact 6th-order and 8th-order linear and nonlinear reconstructions can be developed. As analyzed in this paper, the local linear compact reconstruction can achieve a spectral-like resolution at large wavenumber than the well-established compact scheme of Lele with globally coupled flow variables and their derivatives. For nonlinear gas dynamic evolution, in order to avoid spurious oscillation in discontinuous region, the above compact linear reconstruction from the symmetric stencil can be divided into sub-stencils and apply a biased nonlinear WENO-Z reconstruction. Consequently discontinuous solutions can be captured through the 6th-order and 8th-order compact WENO-type nonlinear reconstruction. In GKS, the time evolution solution of the gas distribution function at a cell interface is based on an integral solution of the kinetic model equation, which covers a physical process from an initial non-equilibrium state to a final equilibrium one. Since the initial non-equilibrium state is obtained based on the nonlinear WENO-Z reconstruction, and the equilibrium state is basically constructed from the linear symmetric reconstruction, the GKS evolution models unifies the nonlinear and linear reconstructions in gas evolution process for the determination of a time-dependent gas distribution function, which gives great advantages in capturing both discontinuous shock wave and the linear aero-acoustic wave in a single computation due to dynamical adaptation of non-equilibrium and equilibrium state from GKS evolution model in different regions. This dynamically adaptive model helps to solve a long lasting problem in the development of high-order schemes about the choices of the linear and nonlinear reconstructions. Compared with discontinuous Galerkin 1 arXiv:1901.00261v1 [physics.comp-ph] 2 Jan 2019 (DG) scheme, the current compact GKS uses the same local and compact stencil, achieves the 6th-order and 8th-order accuracy, uses a much larger time step with CFL number ≥ 0.3, has the robustness as a 2nd-order scheme, and gets accurate solutions in both shock and smooth regions without introducing trouble cell and additional limiting process. The nonlinear reconstruction in the compact scheme is solely based on the WENO-...

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Section: Gas-kinetic Scheme and Multi-stage Multi-derivative Methodsmentioning

confidence: 99%

Section: Multi-stage Multi-derivative Time Stepping Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Compact higher-order gas-kinetic schemes with spectral-like resolution for compressible flow simulations

Zhao

Shyy

et al. 2019

Adv. Aerodyn.

Self Cite

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show abstract

“…It even occupies most of the computation cost in the fourth-order HGKS [14] to calculate the third-order derivatives of particle distribution functions and take their moments. Consequently, the state-of-art HGKS with higher than third-order accuracy is compromised to the multi-stage framework where only low-order HGKS evolution model is utilized [18,9,8].…”

Section: Hgks With Arbitrary High-order Accuracymentioning

confidence: 99%

An efficient high-order gas-kinetic scheme (I): Euler equations

Chen

Jiang

2020

Journal of Computational Physics

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In this paper, an efficient high-order gas-kinetic scheme (EHGKS) is proposed to solve the Euler equations for compressible flows. We re-investigate the underlying mechanism of the high-order gas-kinetic scheme (HGKS) and find a new strategy to improve its efficiency. The main idea of the new scheme contains two parts. Firstly, inspired by the state-of-art simplifications on the third-order HGKS, we extend the HGKS to the case of arbitrary highorder accuracy and eliminate its unnecessary high-order dissipation terms. Secondly, instead of computing the derivatives of particle distribution function and their complex moments, we introduce a Lax-Wendroff procedure to compute the high-order derivatives of macroscopic quantities directly. The new scheme takes advantage of both HGKS and the Lax-Wendroff procedure, so that it can be easily extended to the case of arbitrary high-order accuracy with practical significance. Typical numerical tests are carried out by EHGKS, with the third, fifth and seventh-order accuracy. The presence of good resolution on the discontinuities and flow details, together with the optimal CFL numbers, validates the high accuracy and strong robustness of EHGKS. To compare the efficiency, we present the results computed by the EHGKS, the original HGKS and Runge-Kutta-WENO-GKS. This further demonstrates the advantages of EHGKS.

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Lattice Boltzmann method for compressible Euler equations based on exact kinetic system

Hanada

Kataoka

2021

Numerical Methods in Fluids

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We have developed a new type of simple lattice Boltzmann (LB) model for the compressible Euler equations based on the collisionless kinetic-equation approach proposed by Sone. The model uses the collisionless kinetic equation in the streaming process, and modifies the distribution function to its Chapman-Enskog type at each time step. Compared with the current LB models which solve the kinetic equation of the BGK type, the proposed model is superior in the following two points: (i) Inviscid flows can be computed stably while there is no such model in the current LBM; (ii) The velocity distribution function does not need to be memorized in computation. We calculate various inviscid compressible flows described by the compressible Euler equations using our new one-, two-, and three-dimensional models. Numerical results show the capability of our scheme to simulate high-speed supersonic flows with shock waves.

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A compact fourth-order gas-kinetic scheme for the Euler and Navier–Stokes equations

Cited by 56 publications

References 62 publications

Compact higher-order gas-kinetic schemes with spectral-like resolution for compressible flow simulations

Compact higher-order gas-kinetic schemes with spectral-like resolution for compressible flow simulations

An efficient high-order gas-kinetic scheme (I): Euler equations

Lattice Boltzmann method for compressible Euler equations based on exact kinetic system

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