  In this paper, we are concerned about the lattice Boltzmann methods (LBMs)
based on vector-kinetic models for hyperbolic partial differential equations.
In addition to usual lattice Boltzmann equation (LBE) derived by explicit
discretisation of vector-kinetic equation (VKE), we also consider LBE derived
by semi-implicit discretisation of VKE and compare the relaxation factors of
both. We study the properties such as H-inequality, total variation boundedness
and positivity of both the LBEs, and infer that the LBE due to semi-implicit
discretisation naturally satisfies all the properties while the LBE due to
explicit discretisation requires more restrictive condition on relaxation
factor compared to the usual condition obtained from Chapman-Enskog expansion.
We also derive the macroscopic finite difference form of the LBEs, and utilise
it to establish the consistency of LBEs with the hyperbolic system. Further, we
extend this LBM framework to hyperbolic conservation laws with source terms,
such that there is no spurious numerical convection due to imbalance between
convection and source terms. We also present a D$2$Q$9$ model that allows
upwinding even along diagonal directions in addition to the usual upwinding
along coordinate directions. The different aspects of the results are validated
numerically on standard benchmark problems.
