On the stability and convergence of a semi-discrete discontinuous Galerkin scheme to the kinetic Cucker–Smale model

Abstract

We study analytical properties of a semi-discrete discontinuous Galerkin (DG) scheme for the kinetic Cucker–Smale (CS) equation. The kinetic CS equation appears in the mean-field limit of the particle CS model and it corresponds to the dissipative Vlasov type equation approximating the large particle CS system. For this proposed DG scheme, we show that it exhibits analytical properties such as the conservation of mass, L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-stability and convergence to the sufficiently regular solution, as the mesh-size tends to zero. In particular, we verify that the convergence rate of the DG numerical solution to the sufficiently regular kinetic solution is dependent on the Sobolev regularity of the kinetic soluiton. We also present several numerical simulations for low-dimensional cases.