Physics of Fluids, Volume 32, Issue 12, December 2020. This article presents a new numerical scheme designed to solve for any scalar equation coupled with a lattice Boltzmann solver (in so-called hybrid methods). Its most direct application is solving an energy equation, in parallel with a lattice Boltzmann solver, dealing with mass and momentum conservation. The numerical scheme is specifically designed to compute the energy flux consistently with the mass and momentum flux (as is carried out, for instance, using Riemann solvers). This scheme effectively eliminates a major limitation of the current compressible hybrid lattice Boltzmann method, in which the energy conservation is tackled under a non-conservative form, leading to discretization errors on jump conditions across shocks. Combined with our recently presented pressure-based solver [G. Farag et al., “A pressure-based regularized lattice-Boltzmann method for the simulation of compressible flows,” Phys. Fluids 32(6), 066106 (2020)], the resulting hybrid lattice Boltzmann scheme is, to the authors’ knowledge, the first to numerically conserve mass, momentum, and total energy simultaneously.

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