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A result of convergence for a mono-dimensional two-velocities lattice Boltzmann scheme
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We consider a mono-dimensional two-velocities scheme used to approximate the solutions of a scalar hyperbolic conservative partial differential equation. We prove the convergence of the discrete solution toward the unique entropy solution by first estimating the supremum norm and the total variation of the discrete solution, and second by constructing a discrete kinetic entropy-entropy flux pair being given a continuous entropy-entropy flux pair of the hyperbolic system. We finally illustrate our results with numerical simulations of the advection equation and the Burgers equation.
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A positive- and bound-preserving vectorial lattice Boltzmann method in two dimensions
A blended first- and second-order vectorial lattice Boltzmann scheme with convex limiters provably preserves density and internal energy positivity in two-dimensional compressible Euler simulations.
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