带可逆边界的双弛豫时间向量格子玻尔兹曼格式的网格均匀幂稳定性

中文摘要 AI 辅助

我们研究在均匀静止态附近线性化的向量值双弛豫时间格子玻尔兹曼格式的碰撞-传输算子。假设平衡块为正定，且偶链接和奇链接弛豫参数满足 s₊ + s₋ = 2、0 < s₋ < 2。若齐次传输在平衡度量下为酉算子且在速度交换下可逆，则放大算子的幂次相对于网格节点的数量和排列是一致有界的。可允许的传输包括周期传输、向量半程反弹、坐标对齐的镜面反射、切向正交对合以及兼容的多通道散射。该证明将分布函数方程简化为由压缩算子生成的两步宏观递推。将数值范围包含在椭圆内，结合 Crouzeix-Palencia 定理，可得到伴随算子的与维度无关的估计，进而得到分布函数的界。对于三系数非中点边界插值，我们给出一个精确的有理 D2N5 示例，其有限域放大矩阵存在大于1的实特征值，尽管插值系数和体参数均为可允许的，因此仅系数凸性无法保证该边界族的稳定性。

英文摘要

We consider the collision-transport operator for vector-valued two-relaxation-time lattice Boltzmann schemes linearized about a uniform rest state. Assume that the equilibrium blocks are positive definite and that the link-even and link-odd relaxation parameters satisfy s_+ + s_- = 2 and 0 < s_- < 2. If the homogeneous transport is unitary in the equilibrium metric and reversible under velocity exchange, then the powers of the amplification operator are bounded uniformly with respect to the number and arrangement of lattice nodes. The admissible transports include periodic transport, vector halfway bounce-back, coordinate-aligned specular reflection, tangential orthogonal involutions, and compatible multi-channel scattering. The proof reduces the population equation to a two-step macroscopic recurrence generated by a contraction. An inclusion of the numerical range in an ellipse, combined with the Crouzeix-Palencia theorem, yields a dimension-independent estimate for the companion operator and hence the population bound. For a three-coefficient off-midpoint boundary interpolation, we give an exact rational D2N5 example whose finite-domain amplification matrix has a real eigenvalue larger than one, although the interpolation coefficients and bulk parameters are admissible. Thus coefficient convexity alone does not ensure stability for this boundary family.

发表机构

• Academy for Multidisciplinary Studies, Capital Normal University（首都师范大学多学科研究院）

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