Applying the lattice Boltzmann equation to multiscale fluid problems

phenomena, overstating the need for simulational tools that can handle multiple space and time scales is hard. The capability of addressing problems across several length and time scales is a hallmark of modern computational science, the goal of which is to tackle issues that straddle various traditional disciplines of science and engineering. Some interesting examples of such phenomena that have received considerable attention recently are drug design, the brittle or ductile failure of structural materials, heterogeneous catalysis, and turbulent combustion. We have come to know computational schemes aimed at such complex applications that involve multiple levels of physical and mathematical descriptions as multiscale methods. One approach to multiscale modeling methods is to integrate schemes, which usually apply to a single-scale regime. This approach’s central issue is dealing with the “hand-shaking” regions: At these regions, the different schemes that handle individual scales need to exchange information in a way that is physically meaningful, mathematically consistent, and computationally efficient. Two-level schemes combing atomistic and continuum methods for crack propagation in solids, or strong shock fronts in rarefied gases, made their appearance in the early 1990s.1–3 More recently, researchers have applied threelevel schemes to study crack dynamics, combining a finite-element (FE) treatment of continuum mechanics in regions far from the crack, a molecular dynamics (MD) treatment of atomic motion near the crack, and a quantum mechanical (QM) description of bonding in the crack tip’s immediate neighborhood.4 This FE-MDQM implementation represents a concrete example of composite algorithms—that is, methods that involve seamless interfaces between the different mathematical models associated with different physical levels of description. An alternative approach is to explore methods that can host more than one physical level of description—for example, atomistic, kinetic, and fluid—within the same mathematical framework. A potential candidate is the Lattice Boltzmann equation method. The LBE is a minimal form of the Boltzmann kinetic equation in