一种具有自适应松弛参数和Lax--Friedrichs型平衡态的双曲系统格子玻尔兹曼方法
A Lattice Boltzmann Method with Adaptive Relaxation Parameter and Lax--Friedrichs-Type Equilibrium for Hyperbolic Systems
- University of Science and Technology Beijing（北京科技大学）
- University of Dundee（邓迪大学）
- Capital Normal University（首都师范大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出一种带自适应松弛参数和Lax--Friedrichs型平衡态的格子玻尔兹曼方法，用于带源项双曲系统，在光滑区保持二阶精度、间断处抑制振荡，并通过数值实验验证其精度与稳健性。
AI中文摘要：
本文针对带有源项的双曲系统，提出了一种具有自适应松弛参数和Lax--Friedrichs型平衡态的格子玻尔兹曼方法。该平衡分布恢复守恒变量和物理通量，同时纳入由特征速度界确定的耗散。为平衡精度与鲁棒性，松弛参数基于特征投影的局部光滑性指示器进行选取。在光滑区域，参数接近低耗散极限，保持二阶精度；在间断附近，参数自动减小以引入局部耗散并抑制非物理振荡。该稳定化直接作用于局部碰撞步骤，并保持标准的碰撞-迁移结构，无需后验重算或基于界面的限制。Maxwell迭代确立了光滑区域的二阶一致性，并在标准CFL条件下，对具有周期边界条件的线性双曲系统证明了加权$L^2$稳定性估计。针对标量对流、Euler、浅水和反应Euler方程的数值实验，展示了光滑解的预期精度以及对具有挑战性的一维和二维间断问题（包括干湿边界和反应间断）的稳健分辨能力。二维胞格爆轰的大规模模拟进一步展示了该方法解析长时间多维激波-反应相互作用的能力。
英文摘要：
In this paper, a lattice Boltzmann method with an adaptive relaxation parameter and a Lax--Friedrichs-type equilibrium is proposed for hyperbolic systems with source terms. The equilibrium distribution recovers the conservative variables and physical fluxes while incorporating dissipation determined by characteristic-speed bounds. To balance accuracy and robustness, the relaxation parameter is selected from a local smoothness indicator based on characteristic projections. In smooth regions, the parameter approaches the low-dissipation limit, retaining second-order accuracy; near discontinuities, it is automatically reduced to introduce localized dissipation and suppress nonphysical oscillations. The stabilization acts directly through the local collision step and preserves the standard collide-and-stream structure, without a posteriori recomputation or interface-based limiting. Maxwell iteration establishes second-order consistency in smooth regions, and a weighted $L^2$-stability estimate is proved for linear hyperbolic systems with periodic boundary conditions under a standard CFL condition. Numerical experiments for scalar advection, Euler, shallow-water, and reactive Euler equations demonstrate the expected accuracy for smooth solutions and robust resolution of challenging one- and two-dimensional discontinuous problems, including wet--dry fronts and reactive discontinuities. A large-scale simulation of a 2D cellular detonation further demonstrates the capability of the method to resolve long-time multidimensional shock--reaction interactions.
