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CALIFORNIA STATE UNIVERSITY, NORTHRIDGE Development Of CALIFORNIA STATE UNIVERSITY, NORTHRIDGE Development of Fast Methods for Evaluating the Boltzmann Collision Operator Based on Discontinuous Galerkin Discretizations in the Velocity Variable, Convolution Formulation, and Fast Fourier Transform A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Applied Mathematics By Jeffrey Limbacher August 2018 The thesis of Jeffrey Limbacher is approved: Ali Zakeri , Ph.D. Date Vladislav Panferov , Ph.D. Date Alexander Alekseenko , Ph.D., Chair Date California State University, Northridge ii Acknowledgements My deepest gratitude goes to my advisor, Dr. Alekseenko. Working with him has been an amazing experience. Dr. Alekseenko has given me amazing opportunities, guidance, and support. I would like to thank the CSUN Math department, which has a superb focus on its students. I would like to thank my friends and family for their support throughout my time at CSUN. They have supported my decision to pursue a Master’s degree without hesitation. Lastly, thanks to my girlfriend, Pauline, who has been a constant source of happiness and motivation for me. iii Table of Contents Signature page ii Acknowledgements iii List of Tables vi List of Figures vii Abstract viii 1 Introduction 1 2 The Boltzmann Equation 3 2.1 Binary Collisions of Particles . .4 2.2 The Collision Operator . .5 2.3 Moments of the Distribution Function . .7 2.4 The Maxwellian Distribution . .7 2.5 Dimensionless Reduction . .9 3 Discontinuous Galerkin Discretization in the Velocity Variable 13 3.1 DG Discretization in Velocity Space . 13 3.2 Nodal-DG Velocity Discretization of the Boltzmann Equation . 14 3.3 Reformulation of the Galerkin Projection of the Collision Operator . 15 3.4 Properties of the Kernel A(~v,~v1; φi;j).................. 17 3.5 Rewriting the Collision Operator in the Form of a Convolution . 18 3.6 Discretization of the Collision Integral . 19 3.7 The Micro-Macro Decomposition . 21 4 The Discrete Fourier Transform 23 4.1 The One-Dimensional Discrete Fourier Transform and its Properties . 23 4.1.1 Properties of the DFT . 24 4.1.2 The Convolution Theorem . 25 4.2 Circular Convolution as Linear Convolution . 27 4.2.1 Linear Convolutions and Circular Convolutions . 27 4.2.2 Linear Convolution of Two Finite-Length sequences . 27 4.2.3 Linear Convolution as Circular Convolution . 28 4.2.4 Circular Convolution as Linear Convolution with Aliasing. 30 4.3 The One-Dimensional Fast Fourier Transform . 31 4.3.1 Radix-2 FFT Algorithm . 33 5 Discretization of the Collision Integral and Fast Evaluation of Discrete Con- volution 35 5.1 Formulas for Computing the DFT of the Collision Integral . 35 iv 5.2 The Algorithm and its Complexity . 42 5.3 Periodic Continuation of f and A .................... 44 6 Numerical Results 47 6.1 Reduction in Computational Complexity . 47 6.2 Numerical Results of the Split Form of the Operator . 48 6.3 0d Homogeneuous Relaxation . 52 6.4 Zero-Padding . 54 7 The Model Kinetic Equations and the Rel-ES Method 58 7.1 The BGK Model . 58 7.2 The ES-BGK Model . 59 7.3 Rel-ES . 62 7.4 Experimental Results: 0d Homogeneous Relaxation . 63 8 Hierarchically Semi-Structured Compression of the Kernel A 65 9 Conclusion 69 References 70 v List of Tables 6.1 CPU times for evaluating the collision operator directly and using the Fourier transform. 48 6.2 Absolute errors in conservation of mass and temperature in the discrete collision integral computed using split and non-split formulations. 50 6.3 The Lmax and L1 errors as we increase npad............... 56 6.4 Performance of the method decreases as we increase npad........ 56 vi List of Figures 2.1 Kremer (2010, p. 27), Fig 1.6 . .4 2.2 A 1D Maxwellian distribution. .8 6.1 Evaluation of the collision operator using split and non-split forms: (a) and (d) the split form evaluated using the Fourier transform; (b) and (e) the split form evaluated directly; (c) and (f) the non-split form evaluated using the Fourier transform. 49 R p 6.2 Relaxation of moments fϕi,p = R3 (ui − u¯i) f(t, ~u) du, i = 1, 2, p = 2, 3, 4, 6 in a mix of Maxwellian streams corresponding to a shock wave with Mach number 3.0 obtained by solving the Boltzmann equation using Fourier and direct evaluations of the collision integral. In the case of p = 2, the relaxation of moments is also compared to moments of a DSMC solution [8]. 53 6.3 Relaxation of moments fϕi,p , i = 1, 2, p = 2, 3, 4, 6 in a mix of Maxwellian streams corresponding to a shock wave with Mach number 1.55 ob- tained by solving the Boltzmann equation using Fourier and direct evaluations of the collision integral. 54 7.1 Relaxation of moments fϕi,p , i = 1, 2, p = 2, 3, 4, 6 in a mix of Maxwellian streams corresponding to a shock wave with Mach number 1.55 ob- tained by solving the Boltzmann equation using Fourier and direct evaluations of the collision integral. 64 p 8.1 Color plots of magnitudes of values of A(~v,~v1; φi,jc ) on uniform rect- angle grid. (left) The case of simple numbering. (right) The case of Morton encoding. 66 p 8.2 HSS reduction of a matrix form of A(~v,~v1; φi,jc ) in the Morton encod- ing. Case of 16 × 16 × 16 grid. (left) Geometrical partitioning. (right) Algebraic partitioning. 67 p 8.3 HSS reduction of a matrix form of A(~v,~v1; φi,jc ) in the Morton encod- ing. Case of 32 × 32 × 32 grid. Algebraic partitioning. 67 vii ABSTRACT Development of Fast Methods for Evaluating the Boltzmann Collision Operator Based on Discontinuous Galerkin Discretizations in the Velocity Variable, Convolution Formulation, and Fast Fourier Transform By Jeffrey Limbacher Master of Science in Applied Mathematics Gas flows around high-speed high altitude aircraft and within rocket thrusters contain regions where the particle velocities deviates significantly from the Maxwellian distribution. The gas in these regions is said to be in non-continuum and is best described by kinetic equations. The most physically accurate model is given by the Boltzmann equation. Due to its high computational cost, there is great interest in reducing the computational costs associated with the evaluation of the collision integral. This thesis focuses on reducing the computational costs associated with the collision integral. The Discontinuous Galerkin (DG) method is used to discretize the equation. Discretization leads to a weighted convolution form which costs O(N 8) operations to calculate directly. A discussion of convolutions, their properties, and fast evaluation of convolutions is discussed. A method is introduced based on the Discrete Fourier Transform (DFT) to reduce the calculation of the convolution form down to O(N 6). Empirical results are shown demonstrating that it is conservative and error is minimal. A discussion of two different formulations of the collision integrals are discussed and numerical results are given. A brief discussion of a new approach based on model kinetic equations is also given. viii Chapter 1 Introduction This thesis concerns itself with the evolution of gas flows in low density regimes. This is of particular interest to the engineering community. One example of such a regime is gas flows around high-altitude high-velocity objects flying through the upper atmosphere such as spacecraft and aircraft. Under such regimes, the particles impart a large amount of kinetic energy on the object causing a large transfer of heat to the object. Preventing damage to the object under these circumstances is essential. It is often difficult to replicate these conditions within a laboratory setting. Consequently, there is hope in the development of high fidelity solvers that can simulate the high speed gas flows around these objects to help predict the correct heating patterns. In these gas regimes, the fluid mechanical laws of Navier-Stokes and Fourier break down. In contrast, kinetic theory provide an accurate description by describing particles at the microscopic level. Kinetic theory describes the non-equilibrium dynamics of a gas or any system comprised of a large number of particles. Kinetic equations have found applications in wide range of problems such as rarefied gas dynamics [16] [15], radiative transfer, and semiconductors modeling. The Boltzmann equation is a kinetic equation that describes gases at the molecular level at regimes where Navier-Stokes and Fourier methods fail. Analytic solutions to the Boltzmann equation have been constructed for simple geometries and special molecular potentials. However, the complexity of the equation, along with the the complexities of boundary conditions and gas-to-gas interactions that occur in engineering and physics applications, suggest that only numerical solutions are possible. At the same time, direct numerical computation of the Boltzmann equation remains elusive. The Boltzmann equation is composed of a 1 five-fold integral which must be evaluated at all points in time, space, and velocity resulting in O(n12) computational cost where n is the number of discretization points in space and velocity in each of the three dimensions. This thesis primarily concerns with evaluating the collision integral in velocity space which has of O(n9) operations. This is still computationally prohibitive, there is a strong interest to develop methods that lower the costs. In this thesis, we explore how to speed up evaluation of the collision operator within the Boltzmann equation by using a Discontinuous-Galerkin method based on the work of [17, 1, 2]. The DG method is advantagous for its ability to capture arbitrary surfaces and conserve mass, momentum, and temperature. However, the drawback of this method is its complexity.
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