On the conservativity of the Particles-on-Demand method for solution of the Discrete Boltzmann Equation

Abstract

On the conservativity of the Particles-on-Demand method for the solution of the Discrete Boltzmann Equation It is well known that the standard Lattice-Boltzmann method (LBM) is applicable in the range of small flow velocities and under the isothermal conditions. The novel Particle-on-demand method [1] allows to numerically solve the discrete Boltzmann equation for high Mach numbers. We validate its capabilities with our implementation on the problems with shock waves. In comparison with the standard Lattice Boltzmann Method, the collision step is simple, but the streaming step is implicit, non-conservative and excessively computationally heavy. We propose a way that in specific cases improves the method by making the streaming step explicit and conservative. The results are validated by examining the total mass, momentum and energy change in the problem of shock formation due to the sound wave distortion. The scheme also performs well in both 1D and 3D test Sod problems.

Extensive analysis of the lattice Boltzmann method on shifted stencils

“…These methods rely on off-lattice streaming and polynomial reconstruction of on-lattice populations to match the local velocity and temperature. Given that the latter introduces nonconservative numerical operations, one needs to carefully study and consider effects of the polynomial reconstruction step on the behavior of the scheme [11,15]. To avoid this conservation issue, one could also adapt the shift step by step using only integer-valued shifts that would be chosen, for example, according to the local Ma number or through predefined subdomains in a similar way as for grid refinement patches).…”

“…Mass, momentum, and energy conservation issues were recently reported in Refs. [11,15]. The latter is currently being investigated in more depth, and corresponding results will be presented in a forthcoming paper.…”

“…Deviation of the (eq) xyy moment of (a) the second-order and (b) entropic discrete EDFs from their continuous counterpart as a function of the Mach number (Ma) for a streamwise shift of U x = δ x /δ t and different spanwise Mach numbers: Bottom to top ( ) 0.173, ( ) 0.346, and ( ) 0.519. appropriate shift. This explains why compressible flows can be simulated using low-order velocity discretizations, which are usually restricted in terms of velocity and/or temperature ranges [1][2][3]11,66]. Nevertheless, in order to properly recover the macroscopic behavior of Navier-Stokes-Fourier equations, it is mandatory to have velocity-and temperature-dependent shift velocities which will eventually require (nonconservative) space interpolation.…”

“…Another way to extend the stability domain of the LBM is based on the concept of adaptive stencils [1][2][3]10,11,59]. It is well known that the optimal operation point of the LBM is at velocity zero (as error terms are minimized at this point [7]).…”

Semi-Lagrangian lattice Boltzmann method for compressible flows

“…The physical and numerical parameters can be decoupled in an off-lattice framework, where the velocity set does not need to match the computational grid. In the field of off-lattice Boltzmann methods the semi-Lagrangian lattice Boltzmann method (SLLBM) was recently introduced [24] and further investigated in a number of subsequent works [25][26][27][28][29]. To formulate the streaming step the algorithm follows the trajectories of the lattice Boltzmann equation along their characteristics to find the cells of the corresponding departure points.…”