EULERIAN-LAGRANGIAN RUNGE-KUTTA DISCONTINUOUS GALERKIN METHOD FOR TRANSPORT SIMULATIONS ON UNSTRUCTURED MESHES

The semi-Lagrangian (SL) approach is attractive in transport simulations, e.g., in climate modeling and kinetic models, due to its numerical stability in allowing extra-large time-stepping sizes. For practical problems with complex geometry, schemes on the unstructured meshes are preferred. However, accurate and mass conservative SL methods on unstructured meshes are still under development and encounter several challenges. For instance, when tracking characteristics backward in time, high order curves are required to accurately approximate the shape of upstream cells, which brings in extra computational complexity. To avoid such computational complexity, we propose an Eulerian-Lagrangian Runge-Kutta discontinuous Galerkin method (EL RK DG) in [X. Cai, J.-M. Qiu, and Y. Yang, J. Comput. Phys., 439 (2021), 110392] as an extension of the SL discontinuous Galerkin (DG) methods. This work is a further extension of the algorithm to unstructured triangular meshes with discussion on the treatment of the inflow boundary condition. We also discuss the discrete geometric conservation law. The nonlinear weighted essentially nonoscillatory (WENO) limiter is applied to control oscillations. Desired properties of the proposed method are numerically verified by a set of benchmark tests.