Unexpected convergence of lattice Boltzmann schemes

On rotational invariance of lattice Boltzmann schemes

We propose the derivation of acoustic-type isotropic partial differential equations that are equivalent to linear lattice Boltzmann schemes with a density scalar field and a momentum vector field as conserved moments. The corresponding linear equivalent partial differential equations are generated with a new “Berliner version” of the Taylor expansion method. The details of the implementation ar...

Structural stability of Lattice Boltzmann schemes

The goal of this work is to determine classes of traveling solitary wave solutions for Lattice Boltzmann schemes by means of an hyperbolic ansatz. It is shown that spurious solitary waves can occur in finite-difference solutions of nonlinear wave equation. The occurence of such a spurious solitary wave, which exhibits a very long life time, results in a non-vanishing numerical error for arbitra...

Convergence of Convective-Diffusive Lattice Boltzmann Methods

Lattice Boltzmann methods are numerical schemes derived as a kinetic approximation of an underlying lattice gas. A numerical convergence theory for nonlinear convective-diffusive lattice Boltzmann methods is established. Convergence, consistency, and stability are defined through truncated Hilbert expansions. In this setting it is shown that consistency and stability imply convergence. Monotone...

Implicit Schemes for the Lattice Boltzmann Equation

Towards higher order lattice Boltzmann schemes

In this contribution we extend the Taylor expansion method proposed previously by one of us and establish equivalent partial differential equations of the lattice Boltzmann scheme proposed by d’Humières [11] at an arbitrary order of accuracy. We derive formally the associated dynamical equations for classical thermal and linear fluid models in one to three space dimensions. We use this approach...