Lattice Boltzmann models for high-order partial differential equations
Otomo, Hiroshi.
2019
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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. ... read moreIn 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
\rho^2 and therefore \mathcal(H) doesn't monotonically decrease. Even if
\mathcal(H) increases, however, its increments per a timestep are limited in
\mathcal(O) \left( \epsilon^3 \right) and this fact supports the numerical
stability to some extent.
Thesis (Ph.D.)--Tufts University, 2019.
Submitted to the Dept. of Mathematics.
Advisor: Bruce Boghosian.
Committee: Peter Love, Sauro Succi, and Xiaozhe Hu.
Keywords: Fluid mechanics, Statistical physics, and Computational physics.read less - ID:
- nk322s88n
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