An h-Adaptive RKDG Method for the Two-Dimensional Incompressible Euler Equations and the Guiding Center Vlasov Model

In this paper, we generalize an h-adaptive Runge–Kutta discontinuous Galerkin scheme developed earlier in Zhu et al. (J Sci Comput 69:1346–1365, 2016) for the 1D Vlasov–Poisson system to the guiding center Vlasov model and the 2D time dependent incompressible Euler equations in the vorticity-stream function formulation. The main difficulty of this generalization lies in solving the 2D Poisson equation due to the irregular adaptive mesh with hanging nodes. We adopt a local discontinuous Galerkin method to solve the Poisson equation. The full adaptive algorithm and the related numerical implementation details are included. Extensive numerical tests have been performed to showcase the effectiveness of the adaptive scheme and its advantage over the fixed-mesh scheme in saving computational cost and improving solution quality.