APTT: An accuracy-preserved tensor-train method for the Boltzmann-BGK equation

TL;DR

This work introduces the Accuracy-Preserved Tensor-Train (APTT) method for the Boltzmann-BGK equation, addressing the curse of dimensionality by integrating a Crank-Nicolson–Leap-Frog time discretization with a second-order upwind spatial scheme and compressing the resulting tensor algebra into a low-rank TT representation. A TT-MALS solver then resolves the resulting TT-based linear system efficiently, while careful TT-rounding and RHS construction ensure the discretization accuracy is preserved and mass, momentum, and energy are conserved within prescribed tolerances. The authors provide detailed complexity and convergence analyses, showing per-step costs of order $\mathcal O(m^4 r + m^3 r^2 + m r^6 + m^2 r^4 \#\text{iter})$ and proving error bounds that scale with TT-rounding tolerances; with $\varepsilon_b=\varepsilon_d=(\Delta t)^{1+\varpi}$ and $\varpi$ chosen appropriately (e.g., $\varpi=3$), the method attains second-order accuracy consistent with the underlying discretization. Numerical experiments in 2D and 3D demonstrate substantial speedups and memory savings over GMRES while maintaining the convergence rate and conservation properties, validating APTT as a practical solver for high-dimensional kinetic equations.

Abstract

Solving the Boltzmann-BGK equation with traditional numerical methods suffers from high computational and memory costs due to the curse of dimensionality. In this paper, we propose a novel accuracy-preserved tensor-train (APTT) method to efficiently solve the Boltzmann-BGK equation. A second-order finite difference scheme is applied to discretize the Boltzmann-BGK equation, resulting in a tensor algebraic system at each time step. Based on the low-rank TT representation, the tensor algebraic system is then approximated as a TT-based low-rank system, which is efficiently solved using the TT-modified alternating least-squares (TT-MALS) solver. Thanks to the low-rank TT representation, the APTT method can significantly reduce the computational and memory costs compared to traditional numerical methods. Theoretical analysis demonstrates that the APTT method maintains the same convergence rate as that of the finite difference scheme. The convergence rate and efficiency of the APTT method are validated by several benchmark test cases.

Figures (9)

• Figure 1: Relative errors for the trigonometric initial value problem. Left: two-dimensional case. Right: three-dimensional case.
• Figure 2: The contour plots of the macroscopic variables for the two-dimensional trigonometric initial value problem. The top and bottom rows represent the solutions at $t=0.5$ and $t=1.0$, respectively. The columns, from left to right, correspond to the macroscopic density, velocity on the first axis, and temperature.
• Figure 3: The relaxation to statistical equilibrium problem. Temporal evolution of $f(\bm{x},\bm{v},t)$ in hyper-plane $(\bm{x}, \bm{v}) = (\bm{0}, 0, v_2, v_3)$ and $(\bm{x}, \bm{v}) = (\bm{0}, v_1, v_2, 0)$.
• Figure 4: The relaxation to statistical equilibrium problem. Temporal evolution of density $\rho(\bm{x}, t)$ and velocity $\bm{U}(\bm{x}, t)$.
• Figure 5: The relaxation to statistical equilibrium problem. Temporal evolution of temperature $T(\bm{x}, t)$.

Theorems & Definitions (9)

• Remark 1
• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3: Conservation laws
• proof
• Theorem 4
• proof