Steady-state two-relaxation-time lattice Boltzmann formulation for transport and flow, closed with the compact multi-reflection boundary and interface-conjugate schemes

•The steady-state two-relaxation-time LBM is introduced for flow and transport equations.•The method is stable for highly discontinuous transport coefficients and sources.•The system is closed with the compact multi-reaction MR boundary and interface-conjugate.•The parabolic solutions are exact with the developed Dirichlet, Neumann and Robin MR rules.•Two-phase Stokes flow and heterogeneous linear ADE exemplify the method in grid-rotated slabs. We introduce the steady-state two-relaxation-time (TRT) Lattice Boltzmann method. Owing to the symmetry argument, the bulk system and the closure equations are all expressed in terms of the equilibrium and non-equilibrium unknowns with the half discrete velocity set. The local mass-conservation solvability condition is adjusted to match the stationary, but also the quasi-stationary, solutions of the standard TRT solver. Additionally, the developed compact, boundary and interface-conjugate, multi-reflection (MR) concept preserves the efficient directional bulk structure and shares its parametrization properties. The method is exemplified in grid-inclined stratified slabs for two-phase Stokes flow and the linear advection-diffusion equation featuring the discontinuous coefficients and sources. The piece-wise parabolic benchmark solutions are matched exactly with the novel Dirichlet, pressure-stress, Neumann flux and Robin MR schemes. The popular, anti-bounce-back and shape-fitted Dirichlet continuity schemes are improved in the presence of both interface-parallel and perpendicular advection velocity fields. The steady-state method brings numerous advantages: it skips transient numerical instability, overpasses severe von Neumann parameter range limitations, tolerates high physical contrasts and arbitrary MR coefficients. The method is promising for faster computation of Stokes/Brinkman/Darcy linear flows in heterogeneous soil, but also heat and mass transfer problems governed by an accurate boundary and interface treatment.