Towards higher order lattice Boltzmann schemes

TL;DR

This paper extends the Taylor expansion method to develop higher order lattice Boltzmann schemes, enabling more accurate simulations of thermal and fluid models by adjusting relaxation parameters for fourth order accuracy.

Contribution

It formally derives equivalent PDEs for DDH lattice Boltzmann schemes at arbitrary orders and applies this to improve accuracy in thermal and fluid simulations.

Findings

Achieved fourth order accuracy for thermal models

Validated schemes through numerical eigenmode computations

Compared results with analytical solutions confirming improved precision

Abstract

In this contribution we extend the Taylor expansion method proposed previously by one of us and establish equivalent partial differential equations of DDH lattice Boltzmann scheme 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 to adjust relaxation parameters in order to enforce fourth order accuracy for thermal model and diffusive relaxation modes of the Stokes problem. We apply the resulting scheme for numerical computation of associated eigenmodes and compare our results with analytical references.