Perfectly Matched Layers and Characteristic Boundaries in Lattice Boltzmann: Accuracy vs Cost

TL;DR

This work revisits perfectly matched layers (PML) in the lattice Boltzmann method (LBM) and couples them with characteristic boundary conditions (CBC) to evaluate accuracy–cost trade-offs for artificial boundaries. It implements PML within LBM using a dampening zone and modifies the collision operator, then compares zero-gradient and CBC-based boundary treatments across one- and two-dimensional benchmarks, analyzing how PML width and damping affect performance. Results show that CBC+PML can yield substantial accuracy gains in multi-dimensional flows at moderate overhead when parameters are properly tuned, while ZG+PML remains appealing for lower-accuracy requirements or simpler models. The findings provide practical guidelines for boundary treatment in LBM and motivate data-driven optimization of PML and CBC parameters for improved stability and efficiency.

Abstract

Artificial boundary conditions (BCs) play a ubiquitous role in numerical simulations of transport phenomena in several diverse fields, such as fluid dynamics, electromagnetism, acoustics, geophysics, and many more. They are essential for accurately capturing the behavior of physical systems whenever the simulation domain is truncated for computational efficiency purposes. Ideally, an artificial BC would allow relevant information to enter or leave the computational domain without introducing artifacts or unphysical effects. Boundary conditions designed to control spurious wave reflections are referred to as nonreflective boundary conditions (NRBCs). Another approach is given by the perfectly matched layers (PMLs), in which the computational domain is extended with multiple dampening layers, where outgoing waves are absorbed exponentially in time. In this work, the definition of PML is revised in the context of the lattice Boltzmann method. The impact of adopting different types of BCs at the edge of the dampening zone is evaluated and compared, in terms of both accuracy and computational costs. It is shown that for sufficiently large buffer zones, PMLs allow stable and accurate simulations even when using a simple zeroth-order extrapolation BC. Moreover, employing PMLs in combination with NRBCs potentially offers significant gains in accuracy at a modest computational overhead, provided the parameters of the BC are properly tuned to match the properties of the underlying fluid flow.

Figures (10)

• Figure 1: Discrete velocities for the D2Q9 (left), D2Q17 (center) and D2Q37 (right) velocity stencils.
• Figure 2: Sketch of PML geometry for a right-hand side boundary on an equidistant grid.
• Figure 3: Time evolution of the density $\rho$ and $x$-component of the velocity field $u_x$ for the "density step" test case. Numerical values for the axis extent are $\Delta \rho = 0.0005, \Delta u = 0.001$.
• Figure 4: Time evolution of global errors for density (top row) and velocity (bottom row) for the "density step" benchmark, using ZG BC.
• Figure 5: Time evolution of the global errors for density (left) and velocity (right) fields for the "density step" benchmark, comparing ZG and LODI for the D2Q17 model.