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A multiscale stochastic particle method based on the Fokker-Planck model for nonequilibrium gas flows

JOURNAL OF COMPUTATIONAL PHYSICS 2025年 520卷

作者： Cui, Ziqi Feng, Kaikai Ma, Qihan Zhang, Jun Beihang Univ Sch Aeronaut Sci & Engn 37 Xueyuan Rd Beijing 100191 Peoples R China

The Fokker-Planck (FP) model characterizes the intermolecular collisions as continuous stochastic processes in velocity space. Compared to the Direct Simulation Monte Carlo (DSMC) method, which is grounded in the Boltzmann equation, the stochastic particle method based on the FP model, referred to as the SP-FP method, demonstrates a relative improvement in computational efficiency. However, the traditional SP-FP method, utilizing operator splitting scheme, encounters analogous limitations to DSMC, thereby limiting the application of large temporal-spatial discretization for simulating near-continuum and continuum flows. In this study, two critical advancements are introduced to improve the traditional SP-FP method. First, the exact integration scheme of the ellipsoidal-statistical FP (ESFP) model is derived, enabling the replication of the theoretical solutions for homogeneous relaxation with any time step. Furthermore, leveraging the integration scheme of the ESFP model, we propose a Multiscale stochastic particle (MSP) method. The MSP method quantifies the numerical errors stemming from the operator splitting scheme and corrects them in the relaxation step. This capability allows the MSP method to be employed with larger temporal-spatial discretization for inhomogeneous problems, achieving second-order accuracy while retaining implementation simplicity. Application of the MSP method to various benchmark problems, including Couette flow, Fourier flow, Poiseuille flow, Sod tube flow, Lid-driven cavity flow, and flow through a slit, demonstrates its promise for simulating multiscale nonequilibrium gas flows with high computational efficiency and accuracy.

关键词： Nonequilibrium flow stochastic particle method Fokker-Planck model Multiscale modelling

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An efficient stochastic particle method for moderately high-dimensional nonlinear PDEs

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JOURNAL OF COMPUTATIONAL PHYSICS 2025年 527卷

作者： Lei, Zhengyang Shao, Sihong Xiong, Yunfeng Peking Univ LMAM CAPT Beijing 100871 Peoples R China Peking Univ Sch Math Sci Beijing 100871 Peoples R China Beijing Normal Univ Sch Math Sci Beijing 100875 Peoples R China

Numerical resolution of moderately high-dimensional nonlinear PDEs remains a huge challenge due to the curse of dimensionality for the classical numerical methods including finite difference, finite element and spectral methods. Starting from the weak formulation of the Lawson-Euler scheme, this paper proposes a stochastic particle method (SPM) by tracking the deterministic motion, random jump, resampling and reweighting of particles. Real-valued weighted particles are adopted by SPM to approximate the high-dimensional solution, which automatically adjusts the point distribution to intimate the relevant feature of the solution. A piecewise constant reconstruction with virtual uniform grid is employed to evaluate the nonlinear terms, which fully exploits the intrinsic adaptive characteristic of SPM. Combining both, SPM can achieve the goal of adaptive sampling in time. Numerical experiments on the 6-D Allen-Cahn equation and the 7D Hamiltonian-Jacobi-Bellman equation demonstrate the potential of SPM in solving moderately high-dimensional nonlinear PDEs efficiently while maintaining an acceptable accuracy.

关键词： Curse of dimensionality stochastic particle method Adaptive sampling Nonlinear PDEs Piecewise constant reconstruction Hamiltonian-Jacobi-Bellman equation Allen-Cahn equation Nonlocal model

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A unified stochastic particle method based on the Bhatnagar-Gross-Krook model for polyatomic gases and its combination with DSMC

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JOURNAL OF COMPUTATIONAL PHYSICS 2022年 471卷

作者： Fei, Fei Hu, Yuan Jenny, Patrick Huazhong Univ Sci & Technol Sch Aerosp Engn Wuhan 430074 Peoples R China Swiss Fed Inst Technol Inst Fluid Dynam Sonneggstr 3 CH-8092 Zurich Switzerland Chinese Acad Sci Inst Mech State Key Lab High Temp Gas Dynam Beijing 100190 Peoples R China

Simulating hypersonic flow around a space vehicle is challenging because of the multi -scale and nonequilibrium nature inherent in these flows. To effectively deal with such flows, a hybrid scheme combining the stochastic particle Bhatnagar-Gross-Krook (BGK) method with direct simulation Monte Carlo (DSMC) was developed recently, but only for monatomic gases (Fei et al. (2021) [29]). In this paper, the particle-particle hybrid method is extended to polyatomic gas flows. In the near continuum regime, employing the Ellipsoidal-Statistical BGK model proposed by Dauvois et al. (2021) [22] with discrete levels of vibrational energy, the stochastic particle BGK method for polyatomic gases is first established following the idea of the unified stochastic particle BGK (USP-BGK) scheme. It is proven to be of second-order accuracy in the fluid limit. After that, the USP-BGK scheme with rotational and vibrational energies is combined with DSMC to construct a hybrid scheme. The present hybrid scheme for polyatomic gases is validated with numer-ical tests of homogeneous relaxation, 1D shock structure and 2D hypersonic flows past a wedge and a cylinder. Compared to the other stochastic particle methods, the proposed hybrid scheme can achieve higher accuracy at a much lower computational cost. Therefore, it is a more efficient tool to study multiscale hypersonic flows.(c) 2022 Elsevier Inc. All rights reserved.

关键词： stochastic particle method BGK model DSMC Polyatomic gas Multiscale flows

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Numerical study of a stochastic particle method for homogeneous gas-phase reactions

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2003年第1-3期45卷 329-349页

作者： Kraft, M Wagner, W Univ Cambridge Dept Chem Engn Cambridge CB2 3RA England Waierstrass Inst Appl Anal & Stochast D-10117 Berlin Germany

In this paper, we study a stochastic particle system that describes homogeneous gasphase reactions of a number of chemical species. First, we introduce the system as a Markov jump process and discuss how relevant physical quantities are represented in terms of appropriate random variables. Then, we show how various deterministic equations, used in the literature, are derived from the stochastic system in the limit when the number of particles goes to infinity. Finally, we apply the corresponding stochastic algorithm to a toy problem, a simple formal reaction mechanism, and a real combustion problem. This problem is given by the isothermal combustion of a homogeneous mixture of heptane and air modelled by a detailed reaction mechanism with 107 chemical species and 808 reversible reactions. Heptane as described in this chemical mechanism serves as model-fuel for different types of internal combustion engines. In particular, we study the order of convergence with respect to the number of simulation particles, and illustrate the limitations of the method. (C) 2003 Elsevier Science Ltd. All rights reserved.

关键词： stochastic particle method combustion convergence efficiency

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Ahigh-order unified stochastic particle method based on the Bhatnagar-Gross-Krook model for multi-scale gas flows

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COMPUTER PHYSICS COMMUNICATIONS 2022年 274卷 108303-108303页

作者： Fei, Fei Jenny, Patrick Huazhong Univ Sci & Technol Sch Aerosp Engn Wuhan 430074 Peoples R China Swiss Fed Inst Technol Inst Fluid Dynam Sonneggstr 3 CH-8092 Zurich Switzerland

stochastic particle methods, such as Direct Simulation Monte Carlo (DSMC), are successfully used to simulate rarefied gas flows. However, since traditional stochastic particle methods employ the operator splitting scheme, i.e., decoupling the molecular motion and collisions, they are of low order accuracy in the continuum regime. In contrast, the discrete velocity method can easily construct high order schemes, but its computational and memory cost is quite high due to the huge number of discrete velocity nodes, especially for the high Mach number flows. To improve the efficiency and accuracy of the simulation for multi-scale flows, a high order stochastic particle BGK method was developed in this paper. The DIRK (Diagonally Implicit Runge-Kutta) and WLS-ENO (weighted-least-squares based essentially non-oscillatory) schemes were used to achieve a high order of accuracy in time evolution and space interpolation, respectively. However, the implementation of DIRK and WLS-ENO with computational particles is not straightforward due to the statistical nature of the stochastic particle methods. To circumvent these difficulties, firstly, the micro-macro decomposition of the relaxation term of the BGK model, which was proposed in the unified stochastic particle BGK (USP-BGK) method, was employed. The fluid limit part of the relaxation term was treated following the DIRK scheme, while the kinetic part was solved with an exact integration scheme similar to the traditional stochastic particle BGK (SPBGK) method. Therefore, in the continuum regime, the present stochastic particle method has the same asymptotic preserving and asymptotic accurate properties as the high order semi-Lagrangian method;and in the rarefied regime, it reduces to the traditional SP-BGK method. Additionally, to overcome the effect of thermal noise in the space interpolation, a new discontinuity indicator, which considers the influence of the thermal fluctuation, was also proposed for the WLS-ENO scheme. Se

关键词： High-order accuracy stochastic particle method BGK model Multiscale flows

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Optimal rate of convergence of a stochastic particle method to solutions of 1D viscous scalar conservation laws

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MATHEMATICS OF COMPUTATION 2004年第246期73卷 777-812页

作者： Bossy, M INRIA F-06902 Sophia Antipolis France

This article presents the analysis of the rate of convergence of a stochastic particle method for 1D viscous scalar conservation laws. The convergence rate result is O(Deltat+1/rootN), where N is the number of numerical particles and Deltat is the time step of the first order Euler scheme applied to the dynamic of the interacting particles.

关键词： stochastic particle method viscous scalar conservation laws Euler discretization scheme weak convergence rate

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A unified stochastic particle method with spatiotemporal adaptation for simulating multiscale gas flows

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JOURNAL OF COMPUTATIONAL PHYSICS 2024年 505卷

作者： Feng, Kaikai Cui, Ziqi Tian, Peng Zhang, Jun Beihang Univ Sch Aeronaut Sci & Engn 37 Xueyuan Rd Beijing 100191 Peoples R China

Recently, the Unified stochastic particle (USP) method has emerged as a promising approach for multiscale particle simulations, tailored specifically for non -equilibrium gas flow scenarios. However, in typical simulations of multiscale flow fields, the time step and grid size of the original USP method remain constrained by the smallest spatiotemporal scale of local flow field, limiting its true multiscale simulation potential. To overcome this critical issue, this work introduces a spatiotemporal adaptive USP method. Initially, we analyze the sources of errors in the traditional single -scale stochastic particle methods under large time steps, which leads to the derivation of the governing equations of USP. Building upon this foundation, we propose principles that dictate the rational time step and grid size for an arbitrary USP simulation. Subsequently, we introduce spatiotemporal adaptive approach primarily based on the local gradient length, leading to the development of a USP solver endowed with spatiotemporal adaptability. The universality, reliability, and efficiency of the algorithm have been verified through an examination of three cases: two-dimensional flow over a cylinder, three-dimensional flow over a blunted body, and flow over a scaled X38 -like model. This work forms an integral component of the development initiative for the general USP solver-SPARTACUS-with relevant source code being open -source in an online repository under the GNU General Public License (GPL).

关键词： Rarefied gas dynamics Nonequilibrium gas flows DSMC stochastic particle method Multiscale simulation

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Reduction of the number of particles in the stochastic weighted particle method for the Boltzmann equation

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JOURNAL OF COMPUTATIONAL PHYSICS 1998年第1期145卷 382-405页

作者： Rjasanow, S Schreiber, T Wagner, W Univ Saarland Dept Math D-66041 Saarbrucken Germany Univ Kaiserslautern Dept Informat D-67653 Kaiserslautern Germany Weierstrass Inst Appl Anal & Stochast D-10117 Berlin Germany

Different Ideas for reducing the number of particles in the stochastic weighted particle method for the Boltzmann equation are described and discussed. The corresponding error bounds are obtained and numerical tests for the spatially homogeneous Boltzmann equation presented. It is shown that if an appropriate reduction procedure is used then any effect on the accuracy of the numerical scheme is negligible. (C) 1998 Academic Press.

关键词： Boltzmann equation stochastic particle method random weight transfer collision mechanism numerical experiments

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Simulation of rare events by the stochastic weighted particle method for the Boltzmann equation

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MATHEMATICAL AND COMPUTER MODELLING 2001年第8-9期33卷 907-926页

作者： Rjasanow, S Wagner, W Weierstrass Inst Appl Anal & Stochast D-10117 Berlin Germany Univ Saarland Dept Math D-66041 Saarbrucken Germany

An extension of the stochastic weighted particle method for the numerical treatment of the Boltzmann equation is presented. A new procedure for modelling the inflow boundary condition is introduced and its performance is tested in a two-dimensional example with strong density gradients. A gain factor in computing time of several orders of magnitude is achieved in specific situations. (C) 2001 Elsevier Science Ltd. All rights reserved.

关键词： Boltzmann equation stochastic particle method rare events variance reduction numerical experiments

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A unified stochastic particle Bhatnagar-Gross-Krook method for multiscale gas flows

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JOURNAL OF COMPUTATIONAL PHYSICS 2020年 400卷 108972-000页

作者： Fei, Fei Zhang, Jun Li, Jing Liu, ZhaoHui Huazhong Univ Sci & Technol Sch Aerosp Engn Wuhan 430074 Hubei Peoples R China Beihang Univ Sch Aeronaut Sci & Engn Beijing 100191 Peoples R China Huazhong Univ Sci & Technol Sch Energy & Power Engn State Key Lab Coal Combust Wuhan 430074 Hubei Peoples R China

The stochastic particle method based on Bhatnagar-Gross-Krook (BGK) or ellipsoidal statistical BGK (ESBGK) model approximates the pairwise collisions in the Boltzmann equation using a relaxation process. Therefore, it is more efficient to simulate gas flows at small Knudsen numbers than the counterparts based on the original Boltzmann equation, such as the Direct Simulation Monte Carlo (DSMC) method. However, the traditional stochastic particle BGK method decouples the molecular motions and collisions in analogy to the DSMC method, and hence its transport properties deviate from physical values as the time step size increases. This defect significantly affects its computational accuracy and efficiency for the simulation of multiscale flows, especially when the transport processes in the continuum regime is important. In the present paper, we propose a unified stochastic particle ESBGK (USP-ESBGK) method by combining the molecular convection and collision effects. In the continuum regime, the proposed method can be applied using large temporal-spatial discretization and approaches to the Navier-Stokes solutions with second-order accuracy. Furthermore, it is capable to simulate both the small scale non-equilibrium flows and large scale continuum flows within a unified framework efficiently and accurately. The applications of USP-ESBGK method to a variety of benchmark problems, including Couette flow, thermal Couette flow, Poiseuille flow, Sod tube flow, cavity flow, Taylor-Green vortex, and flow through a slit, demonstrated that it is a promising tool to simulate multiscale gas flows ranging from rarefied to continuum regime. (C) 2019 Elsevier Inc. All rights reserved.

关键词： stochastic particle method BGK model Multiscale modeling Multiscale flows

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