scholarly journals A discontinuous and adaptive reduced order model for the angular discretization of the Boltzmann transport equation

2020 ◽  
Vol 121 (24) ◽  
pp. 5647-5666
Author(s):  
Alexander C. Hughes ◽  
Andrew G. Buchan
Keyword(s):  
Transport Equation ◽  
Boltzmann Transport Equation ◽  
Reduced Order Model ◽  
Boltzmann Transport ◽  
Order Model ◽  
Reduced Order ◽  
Angular Discretization

scholarly journals Efficient Space–Time Reduced Order Model for Linear Dynamical Systems in Python Using Less than 120 Lines of Code

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1690
Author(s):  
Youngkyu Kim ◽  
Karen Wang ◽  
Youngsoo Choi
Keyword(s):  
Dynamical Systems ◽  
Large Scale ◽  
Space Time ◽  
Reduced Order Model ◽  
Boltzmann Transport ◽  
Order Model ◽  
Linear Dynamical Systems ◽  
Reduced Order ◽  
Block Structures ◽  
O 10

A classical reduced order model (ROM) for dynamical problems typically involves only the spatial reduction of a given problem. Recently, a novel space–time ROM for linear dynamical problems has been developed [Choi et al., Space–tume reduced order model for large-scale linear dynamical systems with application to Boltzmann transport problems, Journal of Computational Physics, 2020], which further reduces the problem size by introducing a temporal reduction in addition to a spatial reduction without much loss in accuracy. The authors show an order of a thousand speed-up with a relative error of less than 10−5 for a large-scale Boltzmann transport problem. In this work, we present for the first time the derivation of the space–time least-squares Petrov–Galerkin (LSPG) projection for linear dynamical systems and its corresponding block structures. Utilizing these block structures, we demonstrate the ease of construction of the space–time ROM method with two model problems: 2D diffusion and 2D convection diffusion, with and without a linear source term. For each problem, we demonstrate the entire process of generating the full order model (FOM) data, constructing the space–time ROM, and predicting the reduced-order solutions, all in less than 120 lines of Python code. We compare our LSPG method with the traditional Galerkin method and show that the space–time ROMs can achieve O(10−3) to O(10−4) relative errors for these problems. Depending on parameter–separability, online speed-ups may or may not be achieved. For the FOMs with parameter–separability, the space–time ROMs can achieve O(10) online speed-ups. Finally, we present an error analysis for the space–time LSPG projection and derive an error bound, which shows an improvement compared to traditional spatial Galerkin ROM methods.

Hybrid Ballistic-Diffusive Solution of the Frequency-Dependent Phonon Boltzmann Transport Equation

2016 ◽  
Author(s):  
Pareekshith Allu ◽  
Sandip Mazumder
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Transport Equation ◽  
Knudsen Number ◽  
Spherical Harmonics ◽  
Hybrid Method ◽  
Boltzmann Transport Equation ◽  
Boltzmann Transport ◽  
Partial Differential ◽  
Angular Discretization

The phonon Boltzmann Transport Equation (BTE) is difficult to solve on account of the directional and spectral nature of the phonon intensity, which necessitates angular and spectral discretization, and ultimately results in a large number (typically few hundreds) of four-dimensional partial differential equations. In the ballistic (large Knudsen number) regime, the phonon intensity is highly anisotropic, and therefore, angular resolution is desirable. However, in the diffusive (small Knudsen number) regime, the intensity is fairly isotropic, and hence, angular discretization is wasteful. In such scenarios, the method of spherical harmonics may be effectively used to reduce the large number of directional BTEs to a few partial differential equations. Since the Knudsen number is frequency dependent, the decision to preserve or eliminate angular discretization may be made frequency by frequency based on whether the spectral Knudsen number is large or small. In this article, a hybrid method is proposed in which for some frequency intervals (bands), full angular discretization is used, while for others, the first order spherical harmonics (P1) is invoked to reduce the number of directional BTEs. The accuracy and efficiency of the hybrid method is tested by solving several steady state and transient nanoscale heat conduction problems in two and three-dimensional geometries. Silicon is used as the candidate material. It is found that hybridization is effective in significantly improving the efficiency of solution of the BTE — sometimes by a factor of three — without significant penalty on the accuracy.

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