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

1. Problem Statement

The paper addresses the numerical analysis of the spatially homogeneous Boltzmann equation with angular cutoff for two specific interaction potentials: Maxwellian molecules ($\gamma=0$) and hard potentials ($0 < \gamma \le 1$).

The core challenge lies in the fundamental mismatch between the physical domain and the mathematical requirements of spectral methods:

• Physical Reality: The Boltzmann equation is defined on the unbounded velocity space $\mathbb{R}^3$.
• Numerical Requirement: Fourier spectral methods require a bounded, periodic domain (a torus $D_L = [-L, L]^3$).

Previous approaches often formulated the problem directly on the torus, leading to convergence toward a periodic equilibrium that differs from the physical whole-space Maxwellian equilibrium. Other rigorous approaches (e.g., for the Landau equation) utilized complex "two-step" frameworks involving intermediate truncated partial differential equations (PDEs) to bridge the gap between $\mathbb{R}^3$ and the torus, which introduced significant analytical complexity.

The authors aim to:

1. Develop a novel Fourier spectral method tailored to these potentials.
2. Derive the first rigorous error estimates for such schemes that compare the numerical solution directly to the exact whole-space solution.
3. Validate the theory with numerical simulations.

2. Methodology

The authors propose a refined, direct strategy that avoids constructing complex intermediate truncated PDEs. Instead, they utilize the truncated whole-space exact solution ($f^R = f \psi_R$) as the continuous reference function on the torus.

Key Technical Innovations:

• Truncation and Periodization:

  • A smooth truncation function $\psi_R$ localizes the solution to a ball $B(0, R)$.
  • The domain is set as a torus $D_L$ with $L = \frac{3+\sqrt{2}}{2}R$, ensuring that the support of $\psi_R$ fits strictly within the periodic domain without aliasing the collision operator.
  • The numerical solution $f_N^R$ is defined on the torus, and the global numerical approximation $f_N$ is obtained by zero-extension outside $D_L$.

• Novel Analytical Tools on the Torus:

  • Smoothing Projection Operator ($\tilde{P}_N$): Unlike standard orthogonal projections, the authors use a smoothing projection with a mode-truncation function $\Phi$. This allows for better control of commutators and avoids Gibbs phenomena near the truncation boundary.
  • Periodic Weight Function ($m_k$): A smooth, strictly periodic weight function is constructed that coincides with the standard polynomial weight $\langle v \rangle^k$ inside $B(0, R)$ but remains smooth and periodic on the torus. This is crucial for performing weighted norm estimates on the torus.

• Error Decomposition:
  The proof of convergence relies on an energy method applied to the error function $g = f_N^R - f^R$. The error terms are decomposed into three main components:

  1. $I_1$ (Projection/Discretization Error): Arises from the difference between the discrete collision operator and the continuous one. The authors exploit the structure of the smoothing projection and the commutator estimates between $\tilde{P}_N$ and $m_k$.
  2. $I_2$ (Non-linearity/Consistency Error): Arises from the bilinear nature of the collision operator. The authors use intrinsic bounds for the periodic collision operator $Q_p$ derived from its discrete Fourier convolution structure, bypassing the need for pre/post-collisional variable changes which are invalid on a torus.
  3. $I_3$ (Truncation Error): Arises from the difference between the truncated exact solution and the full exact solution. This is controlled by the decay properties of the exact solution.

3. Key Contributions

1. First Rigorous Error Estimates: The paper provides the first mathematically justified error bounds for a Fourier spectral method applied to the cutoff Boltzmann equation that converge to the whole-space solution, not just a periodic approximation.
2. Direct Strategy: The authors eliminate the need for the complex intermediate truncated PDEs used in previous Landau equation analyses, streamlining the proof significantly.
3. Decoupling of Errors: The analysis cleanly separates the total error into:
   • Spectral Approximation Error: $O(R^{3/2+\epsilon}/N)$.
   • Velocity Truncation Error: $O(R^{-l})$ (polynomial decay) or $O(e^{-c'R^s})$ (exponential decay).
4. Intrinsic Periodic Bounds: The derivation of bounds for the periodic collision operator $Q_p$ using Fourier convolution structures, which are shown to match classical whole-space estimates for compactly supported functions.

4. Main Results (Theorem 1.1)

The main theorem establishes the convergence rate of the numerical solution $f_N$ to the exact solution $f$ in weighted $L^2$ norms.

• Case (i) - Polynomial Decay: If the initial data has polynomial decay, the error bound is:
  $$\|f_N(t) - f(t)\|_{L^2_k} \le C \left( \frac{R^{3/2+\epsilon}}{N} + R^{-l} \right) e^{\kappa t}$$
  This implies that for a fixed time $T$, one can choose $R$ and $N$ to achieve any desired accuracy. The optimal $R$ balances the spectral error (which grows with $R$) and the truncation error (which decays with $R$).

• Case (ii) - Exponential Decay: If the initial data exhibits exponential decay ($f_0 e^{c\langle v \rangle^s} \in L^1$), the convergence is significantly improved:
  $$\|f_N(t) - f(t)\|_{L^2_k} \le C' \left( \frac{R^{3/2+\epsilon}}{N} + e^{-c' R^s} \right) e^{\kappa t}$$
  In this case, the required growth of the truncation radius $R(t)$ is polynomial in time (rather than exponential), and the truncation error decays exponentially.

5. Numerical Simulations

The authors validate their theory through two sets of simulations:

1. Maxwellian Molecules ($\gamma=0$):
   • Compared against the explicit BKW (Bobylev-Krook-Wu) solution.
   • Results: Demonstrated the theoretical trade-off between spectral error and truncation error. For small $N$, spectral error dominates; for large $N$, truncation error dominates. An optimal $L$ (domain size) exists for a given $N$, confirming the theoretical prediction.
2. Hard Sphere Molecules ($\gamma=1$):
   • Simulated the relaxation of a sum of two Gaussians to equilibrium.
   • Results: Observed the approach to equilibrium via relative entropy, Fisher information, and high-order moments.
   • Conservation: Noted that mass and energy are not strictly conserved due to domain truncation but decay slowly as $L$ increases. The numerical solution correctly captures the dynamics, with larger domains yielding better conservation properties.

6. Significance

This work bridges a critical gap between theoretical kinetic theory and practical computational science. By providing a rigorous error analysis for a spectral method on the whole space, it offers a reliable framework for simulating rarefied gas dynamics and plasma physics where analytical solutions are unavailable. The methodology of using smoothing projections and periodic weights to handle the unbounded domain is a significant advancement that could be adapted to other kinetic equations (e.g., Landau, Vlasov-Poisson) where spectral methods are preferred for their high accuracy and noise-free nature compared to Monte Carlo methods.