Fully-Lagrangian and Lattice-Boltzmann Methods for Solving Systems of Conservation Equations

Abstract

A class of "fully-Lagrangian" methods for solving systems of conservation equations is defined. The key step in formulating these methods is the definition of a new set of field variables for which Lagrangian discretization is trivial. Recently popular lattice-Boltzmann simulation schemes for solving such systems are shown to be a useful sub-class of these fully-Lagrangian methods in which (a) the conservation laws are satisfied at each grid point, (b) the Lagrangian variables are expanded perturbatively, and (c) discretization error is used to represent physics. Such schemes are typically derived using methods of kinetic theory. Our numerical analysis approach shows that the conventional physical derivation, while certainly valid and fruitful, is not essential, that it often confuses physics and numerics and that it can be unnecessarily constraining. For example, we show that lattice-Boltzmann-like methods can be non-perturbative and can be made higher-order, implicit and/or with non-uniform grids. Furthermore, our approach provides new perspective on the relationship between lattice-Boltzmann methods and finite-difference techniques. Among other things, we show that the lattice-Boltzmann schemes are only conditionally consistent and in some cases are identical to the well-known Dufort-Frankel method. Through this connection, the lattice-Boltzmann method provides a rational basis for understanding Dufort-Frankel and gives a pathway for its generalization. At the same time, that Dufort Frankel is no longer much used suggests that the lattice-Boltzmann approach might also share this fate.