Semi-Lagrangian Discontinuous Galerkin Methods for Fluid and Kinetic Applications

4:30PM at REC 113
Prof. Jingmei Qiu, University of Delaware
Semi-Lagrangian Discontinuous Galerkin Methods for Fluid and Kinetic Applications
Xiangxiong Zhang

The semi-Lagrangian (SL) scheme for transport problems gains more and more popularity in the computational science community due to its attractive properties. For example, the SL scheme, compared with the Eulerian approach, allows extra large time step evolution by incorporating characteristics tracing mechanism, hence achieving great computational efficiency. In this talk, we introduce a family of high order SL methods coupled with the finite element discontinuous Galerkin (DG) method. The proposed SLDG method is locally mass conservative, highly accurate, free of operator splitting errors, and allows for extra large time stepping sizes with numerical stability and robustness.

For fluid problems, such as linear convection-diffusion, we propose to apply the SLDG [Guo, Nair and Qiu, MWR, 2014] method to the convection term, together with the LDG discretization of the diffusion term coupled with diagonally implicit RK (DIRK) time discretization along characteristics. For the nonlinear incompressible Navier-Stokes equation, backward characteristics tracing with high order accuracy could be challenging. We propose to apply the RK exponential integrator [Celledoni and Comet, JSC, 2009], to frozen the nonlinear advection coefficients and to couple with implicit treatment of linear diffusion terms. Our proposed schemes are mass conservative, truly multi-dimensional without dimensional splitting errors, genuinely high order accurate in both space and time, and highly efficient by allowing extra large time stepping size. The method has been extensively tested and benchmarked with classical test problems for transport, Vlasov models in plasma physics and incompressible Euler and Navior-Stokes system.