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Register a new userNonequilibrium entropy limiters in lattice Boltzmann methods
محدودات الإنحدار الغير المتوازن في طرق البولتزمان الشبكية
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نحن نبني نظاما من محدودي الإنزيم غير التعادل لطرق البولتزمان الشبكية (LBM). هذه المحدودين تمسح الاهتزازات غير الصحيحة بدون تباين الصدمات، ولا تؤثر على الحلول الناعمة. عموما، يقومون بنفس العمل لLBM مثل المحدودين الجريان لطرق الاختلافات المحدودة، والحجوم المحدودة وطرق العناصر المحدودة، ولكن لLBM فكرة أساسية وراء بناء برامج محدودي الإنزيم غير التعادل هي تحويل مجال لكمية مفردة - الإنزيم غير التعادل. هناك مجموعتان من المحدودين: (i) المستندة إلى تقييد الإنزيم غير التعادل (قص الإنزيم) و(ii) المستندة إلى تصفية الإنزيم غير التعادل (تصفية الإنزيم). وتوفر الخصائص الفيزيائية لLBM بعض الفوائد الإضافية: السيطرة على إنتاج الإنزيم وتقدير دقيق للتصحيح الصناعي المدخل. وتم اختبار المحدودين المبنيين على الأمثلة الرياضية التقليدية: أنبوبات الصدمة غير الحرارية بمعدل الكثافة الأولي 1:2 والحجرة المضغوطة المدحلة بشكل ثنائي الأبعاد لأرقام المكثفة Re بين 2000 و 7500 على شبكة كثيفة 100 * 100. وتنطبق جميع بناء المحدودين على التوازنات الكافية وغير الكافية.
We construct a system of nonequilibrium entropy limiters for the lattice Boltzmann methods (LBM). These limiters erase spurious oscillations without blurring of shocks, and do not affect smooth solutions. In general, they do the same work for LBM as flux limiters do for finite differences, finite volumes and finite elements methods, but for LBM the main idea behind the construction of nonequilibrium entropy limiter schemes is to transform a field of a scalar quantity - nonequilibrium entropy. There are two families of limiters: (i) based on restriction of nonequilibrium entropy (entropy trimming) and (ii) based on filtering of nonequilibrium entropy (entropy filtering). The physical properties of LBM provide some additional benefits: the control of entropy production and accurate estimate of introduced artificial dissipation are possible. The constructed limiters are tested on classical numerical examples: 1D athermal shock tubes with an initial density ratio 1:2 and the 2D lid-driven cavity for Reynolds numbers Re between 2000 and 7500 on a coarse 100*100 grid. All limiter constructions are applicable for both entropic and non-entropic quasiequilibria.
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We demonstrate how to produce a stable multispeed lattice Boltzmann method (LBM) for a wide range of velocity sets, many of which were previously thought to be intrinsically unstable. We use non-Gauss--Hermitian cubatures. The method operates stably for almost zero viscosity, has second-order accuracy, suppresses typical spurious oscillation (only a modest Gibbs effect is present) and introduces no artificial viscosity. There is almost no computational cost for this innovation. DISCLAIMER: Additional tests and wide discussion of this preprint show that the claimed property of coupled steps: no artificial dissipation and the second-order accuracy of the method are valid only on sufficiently fine grids. For coarse grids the higher-order terms destroy coupling of steps and additional dissipation appears. The equations are true.
We describe how regularization of lattice Boltzmann methods can be achieved by modifying dissipation. Classes of techniques used to try to improve regularization of LBMs include flux limiters, enforcing the exact correct production of entropy and manipulating non-hydrodynamic modes of the system in relaxation. Each of these techniques corresponds to an additional modification of dissipation compared with the standard LBGK model. Using some standard 1D and 2D benchmarks including the shock tube and lid driven cavity, we explore the effectiveness of these classes of methods.
The lattice-Boltzmann method (LBM) and its variants have emerged as promising, computationally efficient and increasingly popular numerical methods for modelling complex fluid flow. However, it is acknowledged that the method can demonstrate numerical instabilities, e.g., in the vicinity of shocks. We propose a simple and novel technique to stabilise the lattice-Boltzmann method by monitoring the difference between microscopic and macroscopic entropy. Populations are returned to their equilibrium states if a threshold value is exceeded. We coin the name Ehrenfests steps for this procedure in homage to the vehicle that we use to introduce the procedure, namely, the Ehrenfests idea of coarse-graining. The one-dimensional shock tube for a compressible isothermal fluid is a standard benchmark test for hydrodynamic codes. We observe that, of all the LBMs considered in the numerical experiment with the one-dimensional shock tube, only the method which includes Ehrenfests steps is capable of suppressing spurious post-shock oscillations.
The Gibbs entropy of a macroscopic classical system is a function of a probability distribution over phase space, i.e., of an ensemble. In contrast, the Boltzmann entropy is a function on phase space, and is thus defined for an individual system. Our aim is to discuss and compare these two notions of entropy, along with the associated ensemblist and individualist views of thermal equilibrium. Using the Gibbsian ensembles for the computation of the Gibbs entropy, the two notions yield the same (leading order) values for the entropy of a macroscopic system in thermal equilibrium. The two approaches do not, however, necessarily agree for non-equilibrium systems. For those, we argue that the Boltzmann entropy is the one that corresponds to thermodynamic entropy, in particular in connection with the second law of thermodynamics. Moreover, we describe the quantum analog of the Boltzmann entropy, and we argue that the individualist (Boltzmannian) concept of equilibrium is supported by the recent works on thermalization of closed quantum systems.
Current implementations of fluctuating lattice Boltzmann equations (FLBE) describe single component fluids. In this paper, a model based on the continuum kinetic Boltzmann equation for describing multicomponent fluids is extended to incorporate the effects of thermal fluctuations. The thus obtained fluctuating Boltzmann equation is first linearized to apply the theory of linear fluctuations, and expressions for the noise covariances are determined by invoking the fluctuation-dissipation theorem (FDT) directly at the kinetic level. Crucial for our analysis is the projection of the Boltzmann equation onto the ortho-normal Hermite basis. By integrating in space and time the fluctuating Boltzmann equation with a discrete number of velocities, the FLBE is obtained for both ideal and non-ideal multicomponent fluids. Numerical simulations are specialized to the case where mean-field interactions are introduced on the lattice, indicating a proper thermalization of the system.
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