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Register a new userFast convergence and asymptotic preserving of the General Synthetic Iterative Scheme
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Recently the general synthetic iteration scheme (GSIS) is proposed to find the steady-state solution of the Boltzmann equation~cite{SuArXiv2019}, where various numerical simulations have shown that (i) the steady-state solution can be found within dozens of iterations at any Knudsen number $K$, and (ii) the solution is accurate even when the spatial cell size in the bulk region is much larger than the molecular mean free path, i.e. Navier-Stokes solutions are recovered at coarse grids. The first property indicates that the error decay rate between two consecutive iterations decreases to zero with $K$, while the second one implies that the GSIS is asymptotically preserving the Navier-Stokes limit. This paper is dedicated to the rigorous proof of both properties.
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Synthetic turbulence models are a useful tool that provide realistic representations of turbulence, necessary to test theoretical results, to serve as background fields in some numerical simulations, and to test analysis tools. Models of 1D and 3D synthetic turbulence previously developed still required large computational resources. A new wavelet-based model of synthetic turbulence, able to produce a field with tunable spectral law, intermittency and anisotropy, is presented here. The rapid algorithm introduced, based on the classic $p$-model of intermittent turbulence, allows to reach a broad spectral range using a modest computational effort. The model has been tested against the standard diagnostics for intermittent turbulence, i.e. the spectral analysis, the scale-dependent statistics of the field increments, and the multifractal analysis, all showing an excellent response.
In this paper, a fast synthetic iterative scheme is developed to accelerate convergence for the implicit DOM based on the stationary phonon BTE. The key innovative point of the present scheme is the introduction of the macroscopic synthetic diffusion equation for the temperature, which is obtained from the zero- and first-order moment equations of the phonon BTE. The synthetic diffusion equation, which is asymptomatically preserving to the Fouriers heat conduction equation in the diffusive regime, contains a term related to the Fouriers law and a term determined by the second-order moment of the distribution function that reflects the non-Fourier heat transfer. The mesoscopic kinetic equation and macroscopic diffusion equations are tightly coupled together, because the diffusion equation provides the temperature for the BTE, while the BTE provides the high-order moment to the diffusion equation to describe the non-Fourier heat transfer. This synthetic iterative scheme strengthens the coupling of all phonons in the phase space to facilitate the fast convergence from the diffusive to ballistic regimes. Typical numerical tests in one-, two-, and three-dimensional problems demonstrate that our scheme can describe the multiscale heat transfer problems accurately and efficiently. For all test cases convergence is reached within one hundred iteration steps, which is one to three orders of magnitude faster than the traditional implicit DOM in the near-diffusive regime.
Computational fluid dynamics is a direct modeling of physical laws in a discretized space. The basic physical laws include the mass, momentum and energy conservations, physically consistent transport process, and similar domain of dependence and influence between the physical reality and the numerical representation. Therefore, a physically soundable numerical scheme must be a compact one which involves the closest neighboring cells within the domain of dependence for the solution update under a CFL number $(sim 1 )$. In the construction of explicit high-order compact scheme, subcell flow distributions or the equivalent degree of freedoms beyond the cell averaged flow variables must be evolved and updated, such as the gradients of the flow variables inside each control volume. The direct modeling of flow evolution under generalized initial condition will be developed in this paper. The direct modeling will provide the updates of flow variables differently on both sides of a cell interface and limit high-order time derivatives of the flux function nonlinearly in case of discontinuity in time, such as a shock wave moving across a cell interface within a time step. The direct modeling unifies the nonlinear limiters in both space for the data reconstruction and time for the time-dependent flux transport. Under the direct modeling framework, as an example, the high-order compact gas-kinetic scheme (GKS) will be constructed. The scheme shows significant improvement in terms of robustness, accuracy, and efficiency in comparison with the previous high-order compact GKS.
Moment expansions are used as model reduction technique in kinetic gas theory to approximate the Boltzmann equation. Rarefied gas models based on so-called moment equations became increasingly popular recently. However, in a seminal paper by Holway [Phys. Fluids 7/6, (1965)] a fundamental restriction on the existence of the expansion was used to explain sub-shock behavior of shock profile solutions obtained by moment equations. Later, Weiss [Phys. Fluids 8/6, (1996)] argued that this restriction does not exist. We will revisit and discuss their findings and explain that both arguments have a correct and incorrect part. While a general convergence restriction for moment expansions does exist, it cannot be attributed to sub-shock solutions. We will also discuss the implications of the restriction and give some numerical evidence for our considerations.
The general characteristics based off-lattice Boltzmann scheme (BKG) proposed by Bardow et~al.(2006), and the discrete unified gas kinetic scheme (DUGKS) are two methods that successfully overcome the time step restriction by the collision time, which is commonly seen in many other kinetic schemes. Basically, the BKG scheme is a time splitting scheme, while the DUGKS is an un-split finite volume scheme. In this work, we first perform a theoretical analysis of the two schemes in the finite volume framework by comparing their numerical flux evaluations. It is found that the effects of collision term are considered in the reconstructions of the cell-interface distribution function in both schemes, which explains why they can overcome the time step restriction and can give accurate results even as the time step is much larger than the collision time. The difference between the two schemes lies in the treatment of the integral of the collision term, in which the Bardows scheme uses the rectangular rule while the DUGKS uses the trapezoidal rule. The performance of the two schemes, i.e., accuracy, stability, and efficiency are then compared by simulating several two dimensional flows, including the unsteady Taylor-Green vortex flow, the steady lid-driven cavity flow, and the laminar boundary layer problem. It is observed that, the DUGKS can give more accurate results than the BKG scheme. Furthermore, the numerical stability of the BKG scheme decreases as the Courant-Friedrichs-Lewy (CFL) number approaches to 1, while the stability of DUGKS is not affected by the CFL number apparently as long as CFL<1. It is also observed that the BKG scheme is about one time faster than the DUGKS scheme with the same computational mesh and time step.
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