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Introduction Time-Step Splitting Transport Collision Operator Deterministic Methods for the Boltzmann Equation A. Narayan and A. Klo¨ckner AM281.1 Convection-Dominated Problems October 15, 2007 A. Narayan and A. Klo¨ckner Deterministic Methods for the Boltzmann Equation Introduction Time-Step Splitting Transport Collision Operator Outline 1 Introduction The Equation Collisions 2 Time-Step Splitting Operator splitting Runge-Kutta 3 Transport Flux-Balance Methods Semi-Lagrangian Methods 4 Collision Operator Discretization Bobylev-Rjasanow’s Integral Transform Method Pareschi-Russo’s Spectral Method 6 Mouhot-Pareschi’s Sub-N Method A. Narayan and A. Klo¨ckner Deterministic Methods for the Boltzmann Equation Introduction Time-Step Splitting Transport Collision Operator Outline 1 Introduction The Equation Collisions 2 Time-Step Splitting Operator splitting Runge-Kutta 3 Transport Flux-Balance Methods Semi-Lagrangian Methods 4 Collision Operator Discretization Bobylev-Rjasanow’s Integral Transform Method Pareschi-Russo’s Spectral Method 6 Mouhot-Pareschi’s Sub-N Method A. Narayan and A. Klo¨ckner Deterministic Methods for the Boltzmann Equation Describes a rarefied gas. Rarefied gas? → Enough space that particles in the same volume element may have differing velocities. Contrast: Continuum Mechanics →All particles in a volume element are moving with constant velocity. (Navier-Stokes Equations) Introduction Time-Step Splitting Transport Collision Operator The Equation The Boltzmann Equation The Boltzmann Equation: Is an equation of Statistical Mechanics. A. Narayan and A. Klo¨ckner Deterministic Methods for the Boltzmann Equation Rarefied gas? → Enough space that particles in the same volume element may have differing velocities. Contrast: Continuum Mechanics →All particles in a volume element are moving with constant velocity. (Navier-Stokes Equations) Introduction Time-Step Splitting Transport Collision Operator The Equation The Boltzmann Equation The Boltzmann Equation: Is an equation of Statistical Mechanics. Describes a rarefied gas. A. Narayan and A. Klo¨ckner Deterministic Methods for the Boltzmann Equation → Enough space that particles in the same volume element may have differing velocities. Contrast: Continuum Mechanics →All particles in a volume element are moving with constant velocity. (Navier-Stokes Equations) Introduction Time-Step Splitting Transport Collision Operator The Equation The Boltzmann Equation The Boltzmann Equation: Is an equation of Statistical Mechanics. Describes a rarefied gas. Rarefied gas? A. Narayan and A. Klo¨ckner Deterministic Methods for the Boltzmann Equation Contrast: Continuum Mechanics →All particles in a volume element are moving with constant velocity. (Navier-Stokes Equations) Introduction Time-Step Splitting Transport Collision Operator The Equation The Boltzmann Equation The Boltzmann Equation: Is an equation of Statistical Mechanics. Describes a rarefied gas. Rarefied gas? → Enough space that particles in the same volume element may have differing velocities. A. Narayan and A. Klo¨ckner Deterministic Methods for the Boltzmann Equation →All particles in a volume element are moving with constant velocity. (Navier-Stokes Equations) Introduction Time-Step Splitting Transport Collision Operator The Equation The Boltzmann Equation The Boltzmann Equation: Is an equation of Statistical Mechanics. Describes a rarefied gas. Rarefied gas? → Enough space that particles in the same volume element may have differing velocities. Contrast: Continuum Mechanics A. Narayan and A. Klo¨ckner Deterministic Methods for the Boltzmann Equation (Navier-Stokes Equations) Introduction Time-Step Splitting Transport Collision Operator The Equation The Boltzmann Equation The Boltzmann Equation: Is an equation of Statistical Mechanics. Describes a rarefied gas. Rarefied gas? → Enough space that particles in the same volume element may have differing velocities. Contrast: Continuum Mechanics →All particles in a volume element are moving with constant velocity. A. Narayan and A. Klo¨ckner Deterministic Methods for the Boltzmann Equation Introduction Time-Step Splitting Transport Collision Operator The Equation The Boltzmann Equation The Boltzmann Equation: Is an equation of Statistical Mechanics. Describes a rarefied gas. Rarefied gas? → Enough space that particles in the same volume element may have differing velocities. Contrast: Continuum Mechanics →All particles in a volume element are moving with constant velocity. (Navier-Stokes Equations) A. Narayan and A. Klo¨ckner Deterministic Methods for the Boltzmann Equation

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Deterministic Methods for the Boltzmann Equation PDF

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Introduction Time-Step Splitting Transport Collision Operator Deterministic Methods for the Boltzmann Equation A. Narayan and A. Klo¨ckner AM281.1 Convection-Dominated Problems October 15, 2007 A. Narayan and A. Klo¨ckner Deterministic Methods for the Boltzmann Equation Introduction Time-Step Splitting Transport Collision Operator Outline 1 Introduction The Equation Collisions 2 Time-Step Splitting Operator splitting Runge-Kutta 3 Transport Flux-Balance Methods Semi-Lagrangian Methods 4 Collision Operator Discretization Bobylev-Rjasanow’s Integral Transform Method Pareschi-Russo’s Spectral Method 6 Mouhot-Pareschi’s Sub-N Method A. Narayan and A. Klo¨ckner Deterministic Methods for the Boltzmann Equation Introduction Time-Step Splitting Transport Collision Operator Outline 1 Introduction The Equation Collisions 2 Time-Step Splitting Operator splitting Runge-Kutta 3 Transport Flux-Balance Methods Semi-Lagrangian Methods 4 Collision Operator Discretization Bobylev-Rjasanow’s Integral Transform Method Pareschi-Russo’s Spectral Method 6 Mouhot-Pareschi’s Sub-N Method A. Narayan and A. Klo¨ckner Deterministic Methods for the Boltzmann Equation Describes a rarefied gas. Rarefied gas? → Enough space that particles in the same volume element may have differing velocities. Contrast: Continuum Mechanics →All particles in a volume element are moving with constant velocity. (Navier-Stokes Equations) Introduction Time-Step Splitting Transport Collision Operator The Equation The Boltzmann Equation The Boltzmann Equation: Is an equation of Statistical Mechanics. A. Narayan and A. Klo¨ckner Deterministic Methods for the Boltzmann Equation Rarefied gas? → Enough space that particles in the same volume element may have differing velocities. Contrast: Continuum Mechanics →All particles in a volume element are moving with constant velocity. (Navier-Stokes Equations) Introduction Time-Step Splitting Transport Collision Operator The Equation The Boltzmann Equation The Boltzmann Equation: Is an equation of Statistical Mechanics. Describes a rarefied gas. A. Narayan and A. Klo¨ckner Deterministic Methods for the Boltzmann Equation → Enough space that particles in the same volume element may have differing velocities. Contrast: Continuum Mechanics →All particles in a volume element are moving with constant velocity. (Navier-Stokes Equations) Introduction Time-Step Splitting Transport Collision Operator The Equation The Boltzmann Equation The Boltzmann Equation: Is an equation of Statistical Mechanics. Describes a rarefied gas. Rarefied gas? A. Narayan and A. Klo¨ckner Deterministic Methods for the Boltzmann Equation Contrast: Continuum Mechanics →All particles in a volume element are moving with constant velocity. (Navier-Stokes Equations) Introduction Time-Step Splitting Transport Collision Operator The Equation The Boltzmann Equation The Boltzmann Equation: Is an equation of Statistical Mechanics. Describes a rarefied gas. Rarefied gas? → Enough space that particles in the same volume element may have differing velocities. A. Narayan and A. Klo¨ckner Deterministic Methods for the Boltzmann Equation →All particles in a volume element are moving with constant velocity. (Navier-Stokes Equations) Introduction Time-Step Splitting Transport Collision Operator The Equation The Boltzmann Equation The Boltzmann Equation: Is an equation of Statistical Mechanics. Describes a rarefied gas. Rarefied gas? → Enough space that particles in the same volume element may have differing velocities. Contrast: Continuum Mechanics A. Narayan and A. Klo¨ckner Deterministic Methods for the Boltzmann Equation (Navier-Stokes Equations) Introduction Time-Step Splitting Transport Collision Operator The Equation The Boltzmann Equation The Boltzmann Equation: Is an equation of Statistical Mechanics. Describes a rarefied gas. Rarefied gas? → Enough space that particles in the same volume element may have differing velocities. Contrast: Continuum Mechanics →All particles in a volume element are moving with constant velocity. A. Narayan and A. Klo¨ckner Deterministic Methods for the Boltzmann Equation Introduction Time-Step Splitting Transport Collision Operator The Equation The Boltzmann Equation The Boltzmann Equation: Is an equation of Statistical Mechanics. Describes a rarefied gas. Rarefied gas? → Enough space that particles in the same volume element may have differing velocities. Contrast: Continuum Mechanics →All particles in a volume element are moving with constant velocity. (Navier-Stokes Equations) A. Narayan and A. Klo¨ckner Deterministic Methods for the Boltzmann Equation

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