Invited Lecture
Semi-Lagrangian Discontinuous Galerkin Methods for Fluid and Kinetic Applications
University of Delaware, Newark, DE, USA
Abstract
We propose Semi-Lagrangian discontinuous Galerkin (SLDG) schemes for convection-diffusion
and convection-relaxation problems with fluid and kinetic applications.
The classical grid-based Eulerian methods, e.g. finite difference (FD) methods,
finite volume (FV) methods and DG methods, can achieve arbitrary spatial and temporal
order of accuracy, yet they suffer quite stringent time stepping size restriction with
explicit time-stepping methods.
In order to be free of the time step constraint, we propose to use the grid-based SL approach,
which propagates information along characteristics, allowing very large CFL numbers and leading
to computational efficiency. Due to it’s efficiency property, SL schemes are widely used in
incompressible flows, plasma physics,
and global multi-tracer transport in atmospheric modeling.
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.
For kinetic problems, such as the BGK equation, we propose to treat the convection
term by the SLDG method, while the relaxation term is evolved with DIRK methods along
characteristics. Our schemes enjoy mass conservation, high order space-time accuracy
and are free of the CFL constraint. Moreover, our proposed schemes possess asymptotic-preserving
property which preserves the asymptotic Euler limit as the Knudsen’s number going to zero.
The performance is showcased by several benchmark problems.
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