A Two-Stage Fourth Order Time-Accurate Discretization for Lax--Wendroff Type Flow Solvers I. Hyperbolic Conservation Laws

Li, Jiequan; Du, Zhifang

doi:10.1137/15m1052512

Cited by 153 publications

(169 citation statements)

References 26 publications

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“…The latter is an example of a consistent entropy stable finite volume numerical scheme whose solutions [ h , m h , E h ] satisfy (2.1)-(2.4) and thus serves as an example to illustrate the abstract theory and rigorous convergence proofs. The second finite volume method is a well-known second order finite volume method based on the generalized Riemann problem, see, e.g., [4,5,7,44,45]. In what follows we will refer to it as the GRP finite volume method.…”

Section: Finite Volume Methodsmentioning

confidence: 99%

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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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Section: Finite Volume Methodsmentioning

confidence: 99%

“…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%

On oscillatory solutions to the complete Euler system

Feireisl

Klingenberg

Kreml

et al. 2020

Journal of Differential Equations

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“…where Σ p is one cell interface of Ω ijk and n is the outer normal direction. A two-stage fourth-order time-accurate discretization was developed for Lax-Wendroff flow solvers with the generalized Riemann problem (GRP) solver [23] and the gas-kinetic scheme (GKS) [28]. Consider the following time-dependent equation…”

Section: High-order Gas-kinetic Schemementioning

confidence: 99%

High-order gas-kinetic scheme with three-dimensional WENO reconstruction for the Euler and Navier-Stokes solutions

Pan

2020

Computers & Fluids

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In this paper, a simple and efficient third-order weighted essentially non-oscillatory (WENO) reconstruction is developed for three-dimensional flows, in which the idea of two-dimensional WENO-AO scheme on unstructured meshes [43] is adopted. In the classical finite volume type WENO schemes, the linear weights for the candidate stencils are obtained by solving linear systems at Gaussian quadrature points of cell interface. For the three-dimensional scheme, such operations at twenty-four Gaussian quadrature points of a hexahedron would reduce the efficiency greatly, especially for the moving-mesh computation. Another drawback of classical WENO schemes is the appearance of negative weights with irregular local topology, which affect the robustness of spatial reconstruction. In such three-dimensional WENO-AO scheme, a simple strategy of selecting big stencil and sub-stencils for reconstruction is proposed. With the reconstructed quadratic polynomial from big stencil and linear polynomials from sub-stencils, the linear weights are chosen as positive numbers with the requirement that their sum equals one and be independent of local mesh topology. With such WENO reconstruction, a high-order gas-kinetic scheme (HGKS) is developed for both threedimensional inviscid and viscous flows. Taken the grid velocity into account, the scheme is extended into the moving-mesh computation as well. Numerical results are provided to illustrate the good performance of such new finite volume WENO schemes. In the future, such WENO reconstruction will be extended to the unstructured meshes.

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“…An isotropic vortex is added to the mean flow with the perturbations in velocities, temperature and no perturbation in entropy S = p/ρ γ [12], which gives…”

Section: D Isotropic Vortex Propagation Problemmentioning

confidence: 99%

“…The periodic boundary condition is adopted. The exact solution is the perturbation propagating with (U, V ) = (1, 1) [12,18]. The test results based on the density are shown in Table 8.…”

Section: D Isotropic Vortex Propagation Problemmentioning

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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A Two-Stage Fourth Order Time-Accurate Discretization for Lax--Wendroff Type Flow Solvers I. Hyperbolic Conservation Laws

Cited by 153 publications

References 26 publications

On oscillatory solutions to the complete Euler system

On oscillatory solutions to the complete Euler system

High-order gas-kinetic scheme with three-dimensional WENO reconstruction for the Euler and Navier-Stokes solutions

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

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