  We present a set of uniform polynomial equations that provides
multidimensional on-lattice higher-order models of the lattice Boltzmann
theory, while keeping compact the number of discrete velocities. As examples,
we explicitly derive two- and three-dimensional on-lattice models applicable to
describing thermal compressible flows of the accuracy levels of the
Navier-Stokes equations with smaller numbers of discrete velocities in
comparison to the existing models. We demonstrate the accuracy and stability of
the three-dimensional model by using the Riemann problem.
