Restoring the conservativity of characteristic-based segregated models: application to the hybrid lattice Boltzmann method

A general methodology is introduced to build conservative numerical models for fluid simulations based on segregated schemes, where mass, momentum, and energy equations are solved by different methods. It is especially designed here for developing new numerical discretizations of the total energy equation and adapted to a thermal coupling with the lattice Boltzmann method (LBM). The proposed methodology is based on a linear equivalence with standard discretizations of the entropy equation, which, as a characteristic variable of the Euler system, allows efficiently decoupling the energy equation with the LBM. To this extent, any LBM scheme is equivalently written under a finite-volume formulation involving fluxes, which are further included in the total energy equation as numerical corrections. The viscous heat production is implicitly considered thanks to the knowledge of the LBM momentum flux. Three models are subsequently derived: a first-order upwind, a Lax–Wendroff, and a third-order Godunov-type schemes. They are assessed on standard academic test cases: a Couette flow, entropy spot and vortex convections, a Sod shock tube, several two-dimensional Riemann problems, and a shock–vortex interaction. Three key features are then exhibited: (1) the models are conservative by construction, recovering correct jump relations across shock waves; (2) the stability and accuracy of entropy modes can be explicitly controlled; and (3) the low dissipation of the LBM for isentropic phenomena is preserved.