Multistep lattice Boltzmann methods: Theory and applications.

Summary: This paper presents a framework for incorporating arbitrary implicit multistep schemes into the lattice Boltzmann method. While the temporal discretization of the lattice Boltzmann equation is usually derived using a second‐order trapezoidal rule, it appears natural to augment the time discretization by using multistep methods. The effect of incorporating multistep methods into the lattice Boltzmann method is studied in terms of accuracy and stability. Numerical tests for the third‐order accurate Adams‐Moulton method and the second‐order backward differentiation formula show that the temporal order of the method can be increased when the stability properties of multistep methods are considered in accordance with the second Dahlquist barrier. A general multistep time integration framework for the lattice Boltzmann method (LBM) is presented. Concretely, a third‐order Adams‐Molton LBM and a second‐order backward differentiation formula LBM are developed and investigated concerning stability, accuracy, and applicability in comparison to the standard trapezoidal rule. It is shown that the order of convergence can be increased by high‐order multistep methods, taking a changed stability region into account. [ABSTRACT FROM AUTHOR]