Abstract
This paper develops and analyses a semi-discrete numerical method for the two-dimensional Vlasov–Stokes system with periodic boundary condition. The method is based on the coupling of the semi-discrete discontinuous Galerkin method for the Vlasov equation with discontinuous Galerkin scheme for the stationary incompressible Stokes equation. The proposed method is both mass and momentum conservative. Since it is difficult to establish non-negativity of the discrete local density, the generalized discrete Stokes operator become non-coercive and indefinite, and under the smallness condition on the discretization parameter, optimal error estimates are established with help of a modified the Stokes projection to deal with the Stokes part and, with the help of a special projection, to tackle the Vlasov part. Finally, numerical experiments based on the dG method combined with a splitting algorithm are performed.
Funding statement: K. Kumar acknowledges the financial support of the University Grants Commission (UGC), Government of India.
Acknowledgements
K. Kumar and H. Hutridurga thank Laurent Desvillettes for introducing them to the fluid-kinetic equations modelling the thin sprays during the Junior Trimester Program on Kinetic Theory organized at the Hausdorff Research Institute for Mathematics, Bonn. K. Kumar and H. Hutridurga thank the Hausdorff Institute of Mathematics, Bonn, for hosting them during the Junior Trimester Program on Kinetic Theory (Summer of 2019) where this work was initiated. The authors acknowledge the valuable comments and suggestions given by two honourable referees which help to improve our manuscript.
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