Compound waves in a thermohydrodynamic lattice BGK scheme using non-perturbative equilibria

– A previously proposed thermohydrodynamic lattice BGK scheme using nonperturbative, or non-polynomial, equilibria is found to simulate continuum equations that differ from the Euler equations of gas dynamics at leading order. Shock tube simulations with this BGK scheme coincide with solutions obtained by solving these different continuum equations with conventional methods. Both sets of solutions contain unphysical compound waves, shocks attached to rarefactions, where the Euler equations contain contact discontinuities. Lattice Boltzmann or lattice BGK equations [1–4] have proved successful at simulating nearly incompressible and isothermal fluid flows, for which they are a viable alternative to conventional numerical methods [5]. However, applications to flows with substantial temperature fluctuations have proved problematic. Many schemes are subject to instabilities, often attributed to the use of polynomial approximations for the equilibrium distributions. These equilibria may become negative, which violates the assumptions of a discrete H-theorem and may permit instability [5–8]. This has spurred recent work on constructing non-polynomial equilibria that remain positive, and often arise from extremising some entropy function [8–10]. Thermohydrodynamic flows are described most simply by the compressible Euler equations expressing conservation of mass, momentum, and energy – the last comprising both macroscopic kinetic energy, and internal kinetic energy or heat. The Euler equations are valid in the limit of high Reynolds number, when heat and momentum transport associated with molecular processes may be neglected. A more refined description requires the Navier-Stokes-Fourier equations that incorporate viscous and thermal diffusion. In this Letter we investigate the thermohydrodynamic lattice BGK scheme proposed by Renda et al. [8], which uses “non-perturbative equilibria” in an attempt to maintain stability at higher Mach numbers. We show that this scheme simulates continuum equations that differ from the intended compressible Euler equations at leading order, due to an incorrect energy flux. Solutions of shock tube problems with this scheme contain unphysical compound (∗) Now at: OCIAM, Mathematical Institute, 24–29 St Giles’, Oxford, OX1 3LB, UK. Email: [email protected]