A Fully Implicit Method for Lattice Boltzmann Equations

Abstract

Existing approaches for solving the lattice Boltzmann equations with finite difference methods are explicit and semi-implicit; both have certain stability constraints on the time step size. In this work, a fully implicit second-order finite difference scheme is developed. We focus on a parallel, highly scalable, Newton-Krylov-RAS algorithm for the solution of a large sparse nonlinear system of equations arising at each time step. Here, RAS is a restricted additive Schwarz preconditioner based on a first-order spatial discretization. We show numerically that by using the fully implicit method the time step size is no longer constrained by the CFL condition, and the Newton-Krylov-RAS algorithm is scalable on a supercomputer with more than ten thousand processors. Moreover, to calculate the steady state solution we investigate an adaptive time stepping strategy. The total compute time required by the implicit method with adaptive time stepping is much smaller than that of an explicit method for several test cases.

Nonlinearly preconditioned constraint-preserving algorithms for subsurface three-phase flow with capillarity

“…Let S b = ∪S β b , β = p w , S w , S g , S o , and S g = S\S b , then based on the strategy (19), we build the following subspace nonlinear system for each i ∈ S b :…”

“…In spite of the popularity of semi-implicit methods, the fully implicit approach enables to solve all the coupled nonlinear equations simultaneously and implicitly with the relaxation of the stability requirement on the time step size, and has been successfully applied to several classes of important applications [17,18,19,20,21,22,23,24,25,26,27]. Thanks to the simultaneous solution approach, it is allowed to account for more physical principles by readily introducing additional contributions to the existing mathematical model.…”

Coupled lattice Boltzmann method for numerical simulations of fully coupled heart and torso bidomain system in electrocardiology

“…Before to prove the main result of proposition, we introduce the main steps of demonstration with applying Chapman-Enskog expansion to distribution functions φ i of LBE (21) in order to recover governing reaction-diusion equation (15). The macroscopic variable Θ is dened in terms of distribution functions as…”

“…Moreover, in order to overcome the limitations of the constraint CFL stability condition, we extend the method to implicit or semi-implicit time schemes, e.g., by using the θ-method (with θ ∈ [0, 1]) or Runge-Kutta methods, coupled with adaptive time stepping strategies, as e.g. in [21] and the references therein. This coupled LBM method will be shown in a forthcoming paper for more general coupled models with realistic complex geometries.…”

An Implicit Block LU-SGS Algorithm-Based Lattice Boltzmann Flux Solver for Simulation of Hypersonic Flows

“…Lee and Lin [35] extended this scheme into the Taylor-Galerkin finite element method (FEM). Huang et al [36] further applied it into a general FDM. Recently, based on the conventional CFD schemes, Chen and Wang [37] developed an implicit block Lower-Upper Symmetric-Gauss-Seidel (LU-SGS) algorithm for solving flow problems with arbitrary grids.…”