High Order Schemes for Transport Problems: Semi-Lagrangian Schemes with Applications to Plasma Physics and Atmospheric Sciences, and Superconvergence

High order schemes for transport gain lots of popularity in scientific computing community due to their superior properties, such as high efficiency and high resolution. In this dissertation, we systematically investigate the efficient high order numerical schemes for solving transport equations.

In the first part, we develop and implement a class of high order semi-Lagrangian (SL) schemes for linear transport equations, which are further applied to the Vlasov simulations and global transport modeling. Compared with Eulerian type schemes, the SL schemes can take arbitrary large time steps without stability issue, leading to improved computational efficiency. For solving the Vlasov-Possion (VP) system, a high order hybrid methodology, which couples discontinuous Galerkin (DG) schemes and finite difference weighted essentially non-oscillatory (WENO) schemes, is proposed in the Strang splitting framework. The hybrid scheme can take advantage of the numerical ingredients in order to attain good numerical performance. Furthermore, an integral deferred correction method is used to correct the splitting error. Then, the proposed SL method for linear transport is extended for solving spherical transport equations. In particular, a SLDG scheme is formulated on the cubed-sphere geometry. A collection of benchmark numerical tests demonstrate reliability and efficiency of the scheme.

In the second part, via classic Fourier approach, we systematically study superconvergence properties of DG schemes. Superconvergence analysis is important for understanding long time behaviors of DG errors. Based on the eigen-structure analysis of the amplification matrix, we can explain that DG errors will not significantly grow for long time simulations. Specifically, the part of the error that grows linearly in time comes from the dispersion and dissipation errors of the physically relevant eigenvalue. Such an error is superconvergent of order $2k+1$, compared with the standard $(k+1)th$ order accuracy. Furthermore, we investigate superconvergence for DG schemes coupled with a Lax-Wendroff (LW) type time discretization. The numerical evidence shows that the original LWDG scheme, does not numerically exhibit superconvergence. In order to restore such properties, we propose a new LWDG scheme by borrowing techniques from a local DG (LDG) scheme. Numerical experiments are presented to verify superconvergence of the newly proposed scheme.