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The algebras of the title are infinite-dimensional graded Lie algebras $L= \bigoplus_{i=1}^{\infty}L_i$, over a field of positive characteristic $p$, that are generated by an element of degree $1$ and an element of degree $p$, and satisfy $[L_i,L_1]=L_{i+1}$ for $i\ge p$. In case $p=2$ such algebras were classified by Caranti and Vaughan-Lee in 2003. We announce an extension of that classification to arbitrary prime characteristic, and prove several major steps in its proof., Comment: Author Accepted Manuscript, 41 pages. arXiv admin note: text overlap with arXiv:1911.00970

Lattice Boltzmann Models (LBM) and Phase Field Models (PFM) are two of the most widespread approaches for the numerical study of multicomponent fluid systems. Both methods have been successfully employed by several authors but, despite their popularity, still remains unclear how to properly compare them and how they perform on the same problem. Here we present a unified framework for the direct (one-to-one) comparison of the multicomponent LBM against the PFM. We provide analytical guidelines on how to compare the Shan-Chen (SC) lattice Boltzmann model for non-ideal multicomponent fluids with a corresponding free energy (FE) lattice Boltzmann model. Then, in order to properly compare the LBM vs. the PFM, we propose a new formulation for the free energy of the Cahn-Hilliard/Navier-Stokes equations. Finally, the LBM model is numerically compared with the corresponding phase field model solved by means of a pseudo-spectral algorithm. This work constitute a first attempt to set the basis for a quantitative comparison between different algorithms for multicomponent fluids. We limit our scope to the few of the most common variants of the two most widespread methodologies, namely the lattice Boltzmann model (SC and FE variants) and the Phase Field Model., Comment: 26 pages, 8 figures, 2 tables, submitted to J. Comp. Phys