A Performance Comparison of Various Bootstrap Methods for Diffusion Processes

In this paper, we compare the finite sample performances of various bootstrap methods for diffusion processes. Though diffusion processes are widely used to analyze stocks, bonds, and many other financial derivatives, they are known to heavily suffer from size distortions of hypothesis tests. While there are many bootstrap methods applicable to diffusion models to reduce such size distortions, their finite sample performances are yet to be investigated. We perform a Monte Carlo simulation comparing the finite sample properties, and our results show that the strong Taylor approximation method produces the best performance, followed by the Hermite expansion method.

A multiple-relaxation-time collision model by Hermite expansion

The Bhatnagar–Gross–Krook (BGK) single-relaxation-time collision model for the Boltzmann equation serves as the foundation of the lattice BGK (LBGK) method developed in recent years. The description of the collision as a uniform relaxation process of the distribution function towards its equilibrium is, in many scenarios, simplistic. Based on a previous series of papers, we present a collision model formulated as independent relaxations of the irreducible components of the Hermite coefficients in the reference frame moving with the fluid. These components, corresponding to the irreducible representation of the rotation group, are the minimum tensor components that can be separately relaxed without violating rotation symmetry. For the 2nd, 3rd and 4th moments, respectively, two, two and three independent relaxation rates can exist, giving rise to the shear and bulk viscosity, thermal diffusivity and some high-order relaxation process not explicitly manifested in the Navier–Stokes-Fourier equations. Using the binomial transform, the Hermite coefficients are evaluated in the absolute frame to avoid the numerical dissipation introduced by interpolation. Extensive numerical verification is also provided. This article is part of the theme issue ‘Progress in mesoscale methods for fluid dynamics simulation’.

Demonstrating SolarPILOT’s Python API Through Heliostat Optimal Aimpoint Strategy Use Case

SolarPILOT is a software package that generates solar field layouts and characterizes the optical performance of concentrating solar power (CSP) tower systems. SolarPILOT was developed by the National Renewable Energy Laboratory (NREL) as a stand-alone desktop application but has also been incorporated into NREL’s1 System Advisor Model (SAM) in a simplified format. Prior means for user interaction with SolarPILOT have included the application’s graphical interface, the SAM routines with limited configurability, and through a built-in scripting language called “LK.” This paper presents a new, full-featured, Python-based application programmable interface (API) for SolarPILOT, which we hereafter refer to as CoPylot. CoPylot provides access to all SolarPILOT’s capabilities to generate and characterize power tower CSP systems seamlessly through Python. Supported capabilities include (i) creating and destroying a model instance with message reporting tools; (ii) accessing and setting any SolarPILOT variable including custom land boundaries for field layouts; (iii) programmatically managing receiver and heliostat objects with varied attributes for systems with multiple receiver or heliostat types; (iv) generating, assigning, and modifying solar field layouts including the ability to set individual heliostat locations, aimpoints, soiling rates, and reflectivity levels; (v) simulating solar field performance; (vi) returning detailed results describing performance of individual heliostats, the aggregate field, and receiver flux distribution; and, (vii) exporting Python-based model instances to multiple file formats. CoPylot enables Python users to perform detailed CSP tower analysis utilizing either the Hermite expansion technique (analytical) or the SolTrace ray-tracing engine. In addition to CoPylot’s functionality, Python users have access to the over 100,000 open-source libraries to develop, analyze, optimize, and visualize power tower CSP research. This enables CSP researchers to perform analysis that was previously not possible through SolarPILOT’s existing interfaces. This paper discusses the capabilities of CoPylot and presents a use case wherein we demonstrate optimal solar field aiming strategies.

Inverse design of mesoscopic models for compressible flow using the Chapman-Enskog analysis

In this paper, based on simplified Boltzmann equation, we explore the inverse-design of mesoscopic models for compressible flow using the Chapman-Enskog analysis. Starting from the single-relaxation-time Boltzmann equation with an additional source term, two model Boltzmann equations for two reduced distribution functions are obtained, each then also having an additional undetermined source term. Under this general framework and using Navier-Stokes-Fourier (NSF) equations as constraints, the structures of the distribution functions are obtained by the leading-order Chapman-Enskog analysis. Next, five basic constraints for the design of the two source terms are obtained in order to recover the NSF system in the continuum limit. These constraints allow for adjustable bulk-to-shear viscosity ratio, Prandtl number as well as a thermal energy source. The specific forms of the two source terms can be determined through proper physical considerations and numerical implementation requirements. By employing the truncated Hermite expansion, one design for the two source terms is proposed. Moreover, three well-known mesoscopic models in the literature are shown to be compatible with these five constraints. In addition, the consistent implementation of boundary conditions is also explored by using the Chapman-Enskog expansion at the NSF order. Finally, based on the higher-order Chapman-Enskog expansion of the distribution functions, we derive the complete analytical expressions for the viscous stress tensor and the heat flux. Some underlying physics can be further explored using the DNS simulation data based on the proposed model.

A general framework for the inverse design of mesoscopic models based on the Boltzmann equation

In this paper, we present a general framework for the inverse-design of mesoscopic models based on the Boltzmann equation. Starting from the single-relaxation-time Boltzmann equation with an additional source term, two model Boltzmann equations for two reduced distribution functions are obtained, each then also having an additional undetermined source term. Under this general framework and using Navier-Stokes-Fourier (NSF) equations as constraints, the structures of the distribution functions are obtained by the leading-order Chapman-Enskog analysis. Next, five basic constraints for the design of the two source terms are obtained in order to recover the Navier-Stokes-Fourier system in the continuum limit. These constraints allow for adjustable bulk-to-shear viscosity ratio, Prandtl number as well as a thermal energy source. The specific forms of the two source terms can be determined through proper physical considerations and numerical implementation requirements. By employing the truncated Hermite expansion, one design for the two source terms is proposed. Moreover, three well-known mesoscopic models in the literature are shown to be compatible with these five constraints. In addition, the consistent implementation of boundary conditions is also explored by using the Chapman-Enskog expansion at the NSF order. Finally, based on the higher-order Chapman-Enskog expansion of the distribution functions, we derive the complete analytical expressions for the viscous stress tensor and the heat flux. Some underlying physics can be further explored under this framework.

Compressibility in lattice Boltzmann on standard stencils: effects of deviation from reference temperature

With growing interest in the simulation of compressible flows using the lattice Boltzmann (LB) method, a number of different approaches have been developed. These methods can be classified as pertaining to one of two major categories: (i) solvers relying on high-order stencils recovering the Navier–Stokes–Fourier equations, and (ii) approaches relying on classical first-neighbour stencils for the compressible Navier–Stokes equations coupled to an additional (LB-based or classical) solver for the energy balance equation. In most cases, the latter relies on a thermal Hermite expansion of the continuous equilibrium distribution function (EDF) to allow for compressibility. Even though recovering the correct equation of state at the Euler level, it has been observed that deviations of local flow temperature from the reference can result in instabilities and/or over-dissipation. The aim of the present study is to evaluate the stability domain of different EDFs, different collision models, with and without the correction terms for the third-order moments. The study is first based on a linear von Neumann analysis. The correction term for the space- and time-discretized equations is derived via a Chapman–Enskog analysis and further corroborated through spectral dispersion–dissipation curves. Finally, a number of numerical simulations are performed to illustrate the proposed theoretical study. This article is part of the theme issue ‘Fluid dynamics, soft matter and complex systems: recent results and new methods’.

A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation

In this paper, a Cauchy problem for the Laplace equation is considered. We develop a modified Tikhonov regularization method based on Hermite expansion to deal with the ill posed-ness of the problem. The regularization parameter is determined by a discrepancy principle. For various smoothness conditions, the solution process of the method is uniform and the convergence rate can be obtained self-adaptively. Numerical tests are also carried out to verify the effectiveness of the method.