Local kinetic sensors for adaptive mesh and algorithm refinement

Abstract

This paper presents novel refinement sensors for the application to adaptive mesh and algorithm refinement (AMAR) with kinetic models, such as discrete velocity and lattice Boltzmann methods. While refinement criteria for AMAR based on macroscopic variables can be replicated in a purely local, and therefore more scalable, way, the main advantage that can be leveraged when working with discrete velocity and lattice Boltzmann methods is the accessibility of information from the one-particle distribution function. With this accessibility, a novel palette of refinement sensors is introduced, allowing for a set of neatly tailored refinement criteria applicable to resolve characteristic flows features in many relevant domains of fluid mechanics, for instance, those emerging in compressible, turbulent, and non-equilibrium flows or non-ideal fluids. After detailed validation, novel refinement sensors are showcased for the application of adaptive mesh refinement (AMR) to a discrete velocity Boltzmann solver for compressible, viscous, and non-equilibrium flows, demonstrating promising results. The proposed sensors establish an accurate, efficient and scalable approach to kinetic simulations with AMAR, offering a valuable tool for studying complex problems in fluid dynamics and paving the way for future extensions to more specific flow problems.

Figures (8)

• Figure 1: Exemplary two-level grid layout with $r_l=2$ for the illustration of level-wise boundary conditions (BCs) and conservative refluxing. The cell coordinates of the coarser level are denoted with ($i$, $j$) and the finer level with ($v$, $w$), respectively strässle2025a-fully-conservative.
• Figure 2: Illustration of the updates and level-wise information exchanges in the subcycling approach. BC refers to the boundary conditions of type physical, synchronization and prolongation, whereas R refers to restriction and CR to the refluxing correction pass. The horizontal axis marks time and the vertical axis displays the level $l$ and $l+1$ with an exemplary refinement ratio of $r_l = 2$. The order of operations is indicated with gray numbers, interpolations in space with gray arrows, and interpolation in space and time with gray dotted arrows, respectively. In case a next finer level $l+2$ exists, the same recursive scheme is applied to step number four and six strässle2025a-fully-conservative.
• Figure 3: Sensors for the Sod shock tube using the total energy split for different resolutions depicted from left to right; $\delta x = L_x/128$, $\delta x = L_x/512$ and $\delta x = L_x/2048$. The legends hold for the whole row. Note that the $y$-axis of the sensors are logarithmic, which is indicated on the respective axis for convenience.
• Figure 4: Sensors for the two-dimensional Riemann problem case No. $3$ using the internal non-translational energy split for a resolution of $\delta x = \delta y = L_x/1024= L_y/1024$. Note that the color scale is logarithmic, which is indicated on the respective axis for convenience.
• Figure 5: Results and level layout after application of adaptive mesh refinement with the local kinetic sensors for the Sod shock tube using the total energy split. The sensor-specific thresholds to trigger the refinement are denoted in each subfigure as $\varepsilon_{\mathrm{kin.}n} = (l_0, l_1)$ for each level, where $l_0$ denotes the threshold on level $l=0$ and $l_1$ denotes the threshold on level $l=1$. The levels are indicated by color on the lower axis of each subfigure, where the resolution on the levels are $\delta x = L_x/256$ on level $0$ (blue), $\delta x = L_x/512$ on level $1$ (gray) and $\delta x = L_x/1024$ on level $2$ (red). The legends hold for the whole figure.