Lattice-Boltzmann Models for High-Order Partial Differential Equations

Abstract

The lattice Boltzmann (LB) method is a characteristic kinetic-based method in the computational fluid dynamics. Besides its successes in the ordinary flow during this several decades, there exist hurdles for applications to complex flow governed by high-order partial differential equations (PDEs). A severe problem is complication of projection from a kinetic equation to a macroscopic equation. In this study, by suppressing motion at low orders, high-order analysis is simplified significantly. Consequently, high-order LB (HOLB) models for various high-order PDEs such as the Burgers', Korteweg-de Vries, and Kuramoto-Sivashinsky (KS) equations are derived in a systematic way. Via comparisons with analytic solutions and previous studies, it is shown that these models have excellent accuracy and consistence with theory.

The chaotic system governed by the KS equation is simulated and analyzed with the HOLB models. Numerical results are consistent with previous studies that used the spectral method. Moreover, using the Chapman-Enskog method, the mechanism of turbulence is analyzed from viewpoints of multi-scale dynamics. It reveals that balance between the sixth and eighth derivative orders, that is influenced by domain size, plays an important role.

We investigate a way to enhance accuracy and robustness of the HOLB models for the KS equation using the Taylor-series expansion method by examining choices of lattice speeds, the relaxation time, and the equilibrium state. As a result, computational costs are saved up to 92% from the original models.

The H-theorem on the KS dynamics is discussed. Previous studies showed monotonic behaviors of a convex functional \mathcal{H}, minimized at the equilibrium state, contribute to robustness of simulation. This study clarifies motion of global \mathcal{H}, that is the Kullback-Leibler form, is approximately proportional to motion of global ρ² and therefore \mathcal{H} doesn't monotonically decrease. Even if \mathcal{H} increases, however, its increments per a timestep are limited in \mathcal{O} ( ε³) and this fact supports the numerical stability to some extent.