Higher-order Multi-dimensional Limiting Strategy for Correction Procedure via Reconstruction

Park, Jin Seok; Chang, Tae Kyu; Kim, Chongam

doi:10.2514/6.2014-0772

Cited by 10 publications

(6 citation statements)

References 36 publications

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“…LLAV shows better convergence characteristics compared to traditional limiters developed for FR, such as Multi-Dimensional Limiting Process (MLP) [54], especially for steady-state cases. Furthermore LLAV is readily adaptable to implicit time-marching schemes.…”

Section: Localized Laplacian Artificial Viscositymentioning

confidence: 99%

“…The oscillations near the shock generally cause numerical instabilities and negative pressures or densities. Several schemes have been investigated in [53,54,55,56,57] to avoid spurious oscillations around shock waves. However these schemes have not been successfully applied to viscous hypersonic flows and are generally only applicable to explicit time stepping.…”

Section: Shock Capturingmentioning

confidence: 99%

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Implicit high-order flux reconstruction solver for high-speed compressible flows

Vandenhoeck

Lani

2019

Computer Physics Communications

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The present paper addresses the development and implementation of the first high-order Flux Reconstruction (FR) solver for high-speed flows within the open-source COOLFluiD (Computational Object-Oriented Libraries for Fluid Dynamics) platform. The resulting solver is fully implicit and able to simulate compressible flow problems governed by either the Euler or the Navier-Stokes equations in two and three dimensions. Furthermore, it can run in parallel on multiple CPU-cores and is designed to handle unstructured grids consisting of both straight and curved edged quadrilateral or hexahedral elements. While most of the implementation relies on state-of-the-art FR algorithms, an improved and more case-independent shock capturing scheme has been developed in order to tackle the first viscous hypersonic simulations using the FR method. Extensive verification of the FR solver has been performed through the use of reproducible benchmark test cases with flow speeds ranging from subsonic to hypersonic, up to Mach 17.6. The obtained results have been favorably compared to those available in literature. Furthermore, so-called "super-accuracy" is retrieved for certain cases when solving the Euler equations. The strengths of the FR solver in terms of computational accuracy per degree of freedom are also illustrated. Finally, the influence of the characterizing parameters of the FR method as well as the the influence of the novel shock capturing scheme on the accuracy of the developed solver is discussed. order of accuracy per degree of freedom, they also provide a lower numerical dissipation [1]. However, high-order methods are more complicated to implement as compared to low-order methods and are often less robust. The fact that robust high-order mesh generators are not readily available has also limited their use for industrial applications [2]. Nonetheless, high-order methods have drawn considerable attention among researchers over the past couple of years due to their higher accuracy and their capability to deal with complex geometries [3]. Furthermore, the generation of high-order curved meshes has become increasingly supported by mesh generators such as the open-source Gmsh [4]. This has resulted in several successful attempts to develop high-order extensions of the existing FV, FD and FE schemes.FD methods have been extended to higher orders in a straightforward manner by widening the stencil. In this regard, Compact Finite Difference [5] schemes have proven to provide higher orders of accuracy. Nonetheless, all FD methods exhibit the same weakness: they are incapable to readily deal with complex geometries, unless utilized within an overset framework which complicates mesh generation [6,7,8]. FV methods, on the other hand, do not have this shortcoming. As a consequence, several high-order extensions have been developed, such as the k-Exact method [9, 10, 11], the Essential Non-Oscillatory (ENO) [12,13,14], and Weighted Essential Non-Oscillatory (WENO) [15,16,17] schemes. These methods do not generally use a compact ste...

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Section: Localized Laplacian Artificial Viscositymentioning

confidence: 99%

Section: Shock Capturingmentioning

confidence: 99%

Implicit high-order flux reconstruction solver for high-speed compressible flows

Vandenhoeck

Lani

2019

Computer Physics Communications

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“…The parameters of these initial pulses are ǫ 1 = 1 × 10 −2 , ǫ 2 = 1 × 10 −3 , ǫ 3 = 4 × 10 The contours of density and pressure perturbations are plotted in Fig. 25. In order to check the result quantitatively, the density and pressure perturbations along y = 0 are compared with the reference solution from the compact eighth-order GKS on a finer uniform mesh [16].…”

Section: Sound Generation By Viscous Shock-vortex Interactionmentioning

confidence: 99%

“…For discontinuous flows, the success of the above compact schemes is limited. The research on identifying the trouble cells and using additional limiting procedure seems unavoidable in DG [21,22,23] and CPR [24,25] methods.…”

Section: Introductionmentioning

confidence: 99%

A compact high-order gas-kinetic scheme on unstructured mesh for acoustic and shock wave computations

Zhao¹,

Ji²,

Shyy³

et al. 2020

Preprint

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Following the development of a third-order compact gas-kinetic scheme (GKS) for the Euler and Navier-Stokes equations (Journal of Computational Physics 410 (2020) 109367), in this paper an even higher-order compact GKS up to sixth order of accuracy will be constructed for the shock and acoustic wave computation on unstructured mesh. The compactness is defined by the physical domain of dependence for an unstructured triangular cell, which may involve the closest neighbors of neighboring cells. The compactness and high-order accuracy of the scheme are coming from the consistency between the high-order initial reconstruction and the high-order gas evolution model under GKS framework. The high-order evolution solution at a cell interface provides not only a time-accurate flux function, but also the time-evolving flow variables. Therefore, the cell-averaged flow variables and their gradients can be explicitly updated at the next time level from the moments of the same time-dependent gas distribution function. Based on the cell averages and cellaveraged derivatives, both linear and nonlinear high-order reconstruction can be obtained for macroscopic flow variables in the evaluation of local equilibrium and non-equilibrium states. The current nonlinear reconstruction is a combination of WENO and ENO methodology, which is specifically suitable for compact GKS on unstructured mesh with a high-order (≥ 4) accuracy. The initial piecewise discontinuous reconstruction is used for the determination non-equilibrium state and an evolved smooth reconstruction for the equilibrium state. The evolution model in gas-kinetic scheme is based on a relaxation process from non-equilibrium to equilibrium state. The time-accurate gas distribution function in GKS provides the Navier-Stokes flux function directly without separating the inviscid and viscous terms, which simplifies the numerical method on unstructured mesh. Based on the timeaccurate flux solver, the two-stage fourth-order time discretisation can be applied to get a fourth-order time-accurate solution with only two stages, which reduces two reconstructions in comparison with the same time-accurate method with Runge-Kutta time stepping. The current high-order GKS can uniformly capture acoustic and shock waves without identifying trouble cells and implementing additional limiting procedure. In addition, the fourth-up to sixth-order compact GKS can use almost the same time step as a second-order shock capturing scheme. The fourth-order GKS on unstructured mesh will be used in the computations 1 from low speed incompressible viscous flow to the high Mach number shock interaction. The accuracy, efficiency, and robustness of the scheme have been validated. The main conclusion of the paper is that beyond the first-order Riemann solver, the use of high-order gas evolution model seems necessary in the development of high-order schemes.

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“…In DG type schemes Cockburn [18] proposed a TVB type limiter, Du Jie [19] developed a WENO type limiter for CPR scheme. Park et al [20] introduced a multidimensional limiter for CPR. And there are other limiters can be found in [21] [22].…”

Section: Introductionmentioning

confidence: 99%

Research on high-order flux reconstruction method with adaptive mesh for shock capturing

Xia

2020

Preprint

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High order schemes have been developed for a quite long time, and many famous schemes arise like Discontinuous Galerkin (DG), Spectral Difference (SD) and WENO schemes, etc. The Flux Reconstruction (FR) scheme proposed by Huynh has attracted the attention of researchers for its simplicity and efficiency. It’s written in differential form and bridges the DG and SD schemes, which can be constructed with a proper choice of parameter. In this paper, realize FR scheme based on the framework of the open source Adaptive Mesh Refinement (AMR) library p4est. To capture shock sharply, the performance of Localized Laplacian Artificial Viscosity (LLAV) and In-cell Piecewise Integrated Solution methods are compared. As an important way of reducing computational cost, AMR technique integrated in p4est is also combined with FR. The performance of the developed code is tested in both one dimensional and two dimensional and get some quite attracting results.

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Higher-order Multi-dimensional Limiting Strategy for Correction Procedure via Reconstruction

Cited by 10 publications

References 36 publications

Implicit high-order flux reconstruction solver for high-speed compressible flows

Implicit high-order flux reconstruction solver for high-speed compressible flows

A compact high-order gas-kinetic scheme on unstructured mesh for acoustic and shock wave computations

Research on high-order flux reconstruction method with adaptive mesh for shock capturing

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