Stability and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for advection-diffusion problems

Haijin Wang Nanjing University Chi-Wang Shu Brown University Qiang Zhang Nanjing University

Numerical Analysis and Scientific Computing mathscidoc:1610.25034

SIAM Journal on Numerical Analysis, 53, 206-227, 2015
The main purpose of this paper is to analyze the stability and error estimates of the local discontinuous Galerkin (LDG) methods coupled with carefully chosen implicit-explicit (IMEX) Runge-Kutta time discretization up to third order accuracy, for solving one-dimensional linear advection-diffusion equations. In the time discretization the advection term is treated explicitly and the diffusion term implicitly. There are three highlights of this work. The first is that we establish an important relationship between the gradient and interface jump of the numerical solution with the independent numerical solution of the gradient in the LDG methods. The second is that, by aid of the aforementioned relationship and the energy method, we show that the IMEX LDG schemes are unconditionally stable for the linear problems, in the sense that the time-step $\dt$ is only required to be upper-bounded by a constant which depends on the ratio of the diffusion and the square of the advection coefficients and is independent of the mesh-size $h$, even though the advection term is treated explicitly. The last is that under this time step condition, we obtain optimal error estimates in both space and time for the third order IMEX Runge-Kutta time-marching coupled with LDG spatial discretization. Numerical experiments are also given to verify the main results.
local discontinuous Galerkin method, implicit-explicit Runge-Kutta time-marching scheme, advection-diffusion equation, stability, error estimate, energy method
[ Download ] [ 2016-10-12 03:57:28 uploaded by chiwangshu ] [ 1097 downloads ] [ 0 comments ] [ Cited by 7 ]
@inproceedings{haijin2015stability,
  title={Stability and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for advection-diffusion problems},
  author={Haijin Wang, Chi-Wang Shu, and Qiang Zhang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161012035728040444152},
  booktitle={SIAM Journal on Numerical Analysis},
  volume={53},
  pages={206-227},
  year={2015},
}
Haijin Wang, Chi-Wang Shu, and Qiang Zhang. Stability and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for advection-diffusion problems. 2015. Vol. 53. In SIAM Journal on Numerical Analysis. pp.206-227. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161012035728040444152.
Please log in for comment!
 
 
Contact us: office-iccm@tsinghua.edu.cn | Copyright Reserved