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A Fourier spectral method for the cutoff Boltzmann equation: Convergence analysis and numerical simulation

Yanzhi Gui, Ling-Bing He, Liu Liu

Abstract

This work addresses a central challenge in the numerical analysis of the cutoff spatially homogeneous Boltzmann equation: the development of rigorously justified, accurate numerical schemes. We present (i) a novel Fourier spectral method for the equation with Maxwellian and hard potentials, (ii) the derivation of the first rigorous error estimates for the proposed schemes. Comprehensive numerical experiments validate the theory, confirming the predicted accuracy and illustrating the method's capability to capture solution dynamics, including the approach to equilibrium. The study thus provides a complete framework--from theoretical analysis to practical implementation--for the reliable computation of solutions to this foundational kinetic model.

A Fourier spectral method for the cutoff Boltzmann equation: Convergence analysis and numerical simulation

Abstract

This work addresses a central challenge in the numerical analysis of the cutoff spatially homogeneous Boltzmann equation: the development of rigorously justified, accurate numerical schemes. We present (i) a novel Fourier spectral method for the equation with Maxwellian and hard potentials, (ii) the derivation of the first rigorous error estimates for the proposed schemes. Comprehensive numerical experiments validate the theory, confirming the predicted accuracy and illustrating the method's capability to capture solution dynamics, including the approach to equilibrium. The study thus provides a complete framework--from theoretical analysis to practical implementation--for the reliable computation of solutions to this foundational kinetic model.

Paper Structure

This paper contains 30 sections, 19 theorems, 275 equations, 5 figures.

Table of Contents

  1. Introduction
  2. A brief review on the Boltzmann equation
  3. Conservation Laws
  4. Entropy Structure and H-Theorem
  5. Existing Analytical Results
  6. Existing Numerical Results
  7. Notations and function spaces
  8. Goals and difficulties.
  9. Fourier-Galerkin spectral methods for the periodic Boltzmann equation
  10. Progress on numerical analysis of spatially homogeneous Landau equation
  11. Strategies and main results.
  12. Main result
  13. Strategy
  14. Organization of the paper
  15. Preliminary results
  16. ...and 15 more sections

Key Result

Proposition 1.1

Suppose that the initial data $0\le f_0\in L^1_{1,0,1}(\mathop{\mathbb R}\nolimits^3)\cap L\log L(\mathop{\mathbb R}\nolimits^3)$ to 1 satisfies The equation 1 admits a unique global solution $f\in L^\infty([0,\infty);H^1_\ell(\mathop{\mathbb R}\nolimits^3))$ with initial data $f_0$ for any positive number $\ell$.

Figures (5)

  • Figure 1: Maxwellian molecules: time evolution of the $L^2$ error in log scale for the scheme \ref{['eq:num']} with respect to $L$ for $2N=32$, $2N=48$ and $2N=64$.
  • Figure 2: Maxwellian molecules: time evolution of the $L^2$ error in log scale for the scheme \ref{['eq:num']} with respect to $N$ for $L=12$, $L=13$ and $L=14$.
  • Figure 3: Hard sphere molecules: Cross-section $f(0,0,v_3)$ at different times $t=0, 0.5, 1, 1.5, 3, 6$.
  • Figure 4: Hard sphere molecules: Time evolution of the mass (left) and energy (right) for $2N=64$ with different domain sizes $L=16, 17, 18$.
  • Figure 5: Hard sphere molecules: Time evolution of entropy $H(f)$ (top left), Fisher information $I(f)$ (top right), relative entropy $H(f|\mu)$ (bottom left), and relative $L^2$ distance $\|f-\mu\|_{L^2}$ (bottom right) for $2N=64$ with different domain sizes $L=16, 17, 18$.

Theorems & Definitions (38)

  • Remark 1.1
  • Definition 1.1: Numerical solutions to \ref{['1']}
  • Remark 1.2
  • Proposition 1.1
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.1
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • ...and 28 more