A velocity-space adaptive unified gas kinetic scheme for continuum and rarefied flows

Xiao, Tianbai; Liu, Chang; Xu, Kun; Cai, Qingchun

doi:10.1016/j.jcp.2020.109535

Cited by 36 publications

(17 citation statements)

References 36 publications

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“…For future work, it is possible to apply the scheme to other complex systems, e.g., astrophysics [56], particle transports [57], and plasma physics [42]. An alternative to a hyperbolicity-preserving limiter is the careful alteration of the SG system itself, such that its hyperbolicity domain is significantly enlarged, or possibly the whole space.…”

Section: Discussionmentioning

confidence: 99%

A flux reconstruction stochastic Galerkin scheme for hyperbolic conservation laws

Xiao¹,

Kusch²,

Koellermeier³

2021

Preprint

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The study of uncertainty propagation poses a great challenge to design numerical solvers with high fidelity. Based on the stochastic Galerkin formulation, this paper addresses the idea and implementation of the first flux reconstruction scheme for hyperbolic conservation laws with random inputs. Unlike the finite volume method, the treatments in physical and random space are consistent, e.g., the modal representation of solutions based on an orthogonal polynomial basis and the nodal representation based on solution collocation points. Therefore, the numerical behaviors of the scheme in the phase space can be designed and understood uniformly. A family of filters is extended to multi-dimensional cases to mitigate the well-known Gibbs phenomenon arising from discontinuities in both physical and random space. The filter function is switched on and off by the dynamic detection of discontinuous solutions, and a slope limiter is employed to preserve the positivity of physically realizable solutions. As a result, the proposed method is able to capture stochastic cross-scale flow evolution where resolved and unresolved regions coexist. Numerical experiments including wave propagation, Burgers' shock, one-dimensional Riemann problem, and two-dimensional shock-vortex interaction problem are presented to validate the scheme. The order of convergence of the current scheme is identified. The capability of the scheme for simulating smooth and discontinuous stochastic flow dynamics is demonstrated. The open-source codes to reproduce the numerical results are available under the MIT license [1].

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Section: Discussionmentioning

confidence: 99%

A flux reconstruction stochastic Galerkin scheme for hyperbolic conservation laws

Xiao¹,

Kusch²,

Koellermeier³

2021

Preprint

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“…Qin et al [143] presented a simple local discrete velocity adaptation in the UGKS under a uniform background velocity mesh, which avoids the interpolation between different levels of velocity grids. Xiao et al [144] replaced the discrete velocity points in the UGKS with a continuous velocity space in the region where the NS solutions provided by the GKS are valid. In the studies of [48,125], the unstructured mesh is adopted in the velocity space for the DUGKS, where more discrete velocity points are arranged in the velocity regions that enclose a large number of molecules, and fewer points are needed in the regions with a small amount of molecules, so that the memory consumption can be reduced and the computational efficiency can be increased.…”

Section: Adaptive Meshmentioning

confidence: 99%

“…With the adaptive wave-particle formulation, the cost of the UGKWP method can be reduced to a particle method in the highly rarefied flow regime, and it becomes a hydrodynamic NS solver in the continuum regime automatically. It is a novel technique that combines and optimizes the adaptive velocity space method [47] and GKS/UGKS hybrid algorithm [144]. Different from the hybrid methods [146,160,161], which are based on the domain decomposition and solver hybridization, the UGKWP method describes the physical state by an adaptive wave-particle decomposition in each cell with a unified treatment in the whole computational domain.…”

Section: Wave-particle Adaptationmentioning

confidence: 99%

GKS and UGKS for High-Speed Flows

Zhu

Zhong

2021

Aerospace

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The gas-kinetic scheme (GKS) and the unified gas-kinetic scheme (UGKS) are numerical methods based on the gas-kinetic theory, which have been widely used in the numerical simulations of high-speed and non-equilibrium flows. Both methods employ a multiscale flux function constructed from the integral solutions of kinetic equations to describe the local evolution process of particles’ free transport and collision. The accumulating effect of particles’ collision during transport process within a time step is used in the construction of the schemes, and the intrinsic simulating flow physics in the schemes depends on the ratio of the particle collision time and the time step, i.e., the so-called cell’s Knudsen number. With the initial distribution function reconstructed from the Chapman–Enskog expansion, the GKS can recover the Navier–Stokes solutions in the continuum regime at a small Knudsen number, and gain multi-dimensional properties by taking into account both normal and tangential flow variations in the flux function. By employing a discrete velocity distribution function, the UGKS can capture highly non-equilibrium physics, and is capable of simulating continuum and rarefied flow in all Knudsen number regimes. For high-speed non-equilibrium flow simulation, the real gas effects should be considered, and the computational efficiency and robustness of the schemes are the great challenges. Therefore, many efforts have been made to improve the validity and reliability of the GKS and UGKS in both the physical modeling and numerical techniques. In this paper, we give a review of the development of the GKS and UGKS in the past decades, such as physical modeling of a diatomic gas with molecular rotation and vibration at high temperature, plasma physics, computational techniques including implicit and multigrid acceleration, memory reduction methods, and wave–particle adaptation.

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“…Here M is the number of grid points for discretizing polar angles, which is much smaller than the number of velocity grids N in each direction. The advantageous efficiency and efficiency guarantee its dominance in wide applications [36,37,[45][46][47]. Here, we briefly introduce the basic idea of this method.…”

Section: Collisionmentioning

confidence: 99%

A flux reconstruction kinetic scheme for the Boltzmann equation

Xiao¹

2021

Preprint

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It is challenging to solve the Boltzmann equation accurately due to the extremely high dimensionality and nonlinearity. This paper addresses the idea and implementation of the first flux reconstruction method for high-order Boltzmann solutions. Based on the Lagrange interpolation and reconstruction, the kinetic upwind flux functions are solved simultaneously within physical and particle velocity space. The fast spectral method is incorporated to solve the full Boltzmann collision integral with a general collision kernel. The explicit singly diagonally implicit Runge-Kutta (ESDIRK) method is employed as time integrator and the stiffness of the collision term is smoothly overcome. Besides, we ensure the shock capturing property by introducing a self-adaptive artificial dissipation, which is derived naturally from the effective cell Knudsen number at the kinetic scale. As a result, the current flux reconstruction kinetic scheme can be universally applied in all flow regimes. Numerical experiments including wave propagation, normal shock structure, one-dimensional Riemann problem, Couette flow and lid-driven cavity will be presented to validate the scheme. The order of convergence of the current scheme is clearly identified. The capability for simulating cross-scale and non-equilibrium flow dynamics is demonstrated.

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A velocity-space adaptive unified gas kinetic scheme for continuum and rarefied flows

Cited by 36 publications

References 36 publications

A flux reconstruction stochastic Galerkin scheme for hyperbolic conservation laws

A flux reconstruction stochastic Galerkin scheme for hyperbolic conservation laws

GKS and UGKS for High-Speed Flows

A flux reconstruction kinetic scheme for the Boltzmann equation

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