Numerical Analysis of Lattice Boltzmann Methods for the Heat Equation on a Bounded Interval: by Jan-Phillip Weiss

Description

The solution of fluid-dynamic equations from the viewpoint of analytical, numerical and algorithmic aspects is a challenging task. In recent years, lattice Boltzmann methods have been developed to face this problem. The performance of the proposed lattice Boltzmann methods renders convincing results, but less is known about the mathematical analysis.

Fluid-dynamic equations, for example the Navier-Stokes or the Euler equations, deal with macroscopic quantities like velocity, pressure or temperature. Lattice Boltzmann methods follow a different approach. They treat distribution functions that stem from a particle kinetic framework. Both approaches rely on conservation principles, the first one on a macroscopic level, the second one on a microscopic level. Since only macroscopic quantities, that is, physically measurable quantities, are in the scope of the researcher's interest, the distribution functions have to be averaged appropriately to achieve interpretable results. With respect to specific scalings, it turns out that the averaged quantities approximate solutions of the fluid-dynamic equations in a certain limit; see for example Ref. [24].

It is an open question whether these different approaches can be related to each other rigorously or if they represent two independent models. Not only owing to the complexity of the limiting equations, the analysis of the lattice Boltzmann methods in the multi-dimensional case is equipped with formidable difficulties. Up to now there are only a few answers concerning essential issues like stability or convergence of the used schemes. But the convincing results achieved so far are encouraging to advance these methods and their analysis.

By the end of the 1980s, lattice Boltzmann methods were introduced by engi- neers and physicists. Many papers have been written since then, but the mathematical background is still obscure in many fields. Areas of application of lattice Boltzmann methods are the simulation of incompressible flows in complex geometries, for example the flow of blood in vessels, multiphase and multicomponent fluids, free surface problems, moving boundaries, fluid-structure interactions, chemical reactions, flow through porous media, suspension flows, magnetohydrodynamics, semiconductor simulations, non-Newtonian fluids, large eddy and turbulence simulations in aerodynamics, and many more.

The limiting fluid-dynamic equations are determined by the scaling and the choice of the collision operator, where various models are possible. For this reason, lattice Boltzmann methods are applicable to a great variety of different problems. Although lattice Boltzmann methods are universally acclaimed for the applicability to complex geometries and interfacial dynamics, severe problems appear in the case of boundary conditions.

The advantages of lattice Boltzmann methods find expression in a comparably simple explicit algorithm on uniform grids with only local interactions. The parallelization of the algorithms for the speed-up of the computations is straightforward. A great advantage is the gain of differentiated quantities without performing numerical differentiations.

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