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The main purpose of this paper is to analyze the stability and error estimates of the local discontinuous Galerkin (LDG) methods coupled with implicit-explicit (IMEX) time discretization schemes, for solving multi-dimensional convection-diffusion equations with nonlinear convection. By establishing the important relationship between the gradient and the interface jump of the numerical solution with the independent numerical solution of the gradient in the LDG method, on both rectangular and triangular elements, we can obtain the same stability results as in the one-dimensional case [H.J. Wang, C.-W. Shu and Q. Zhang, SIAM J. Numer. Anal. 53 (2015) 206–227; H.J. Wang, C.-W. Shu and Q. Zhang, Appl. Math. Comput. 272 (2015) 237–258], i.e., the IMEX LDG schemes are unconditionally stable for the multi-dimensional convection-diffusion problems, in the sense that the time-step is only required to be upper-bounded by a positive constant independent of the spatial mesh size . Furthermore, by the aid of the so-called elliptic projection and the adjoint argument, we can also obtain optimal error estimates in both space and time, for the corresponding fully discrete IMEX LDG schemes, under the same condition, i.e., if piecewise polynomial of degree is adopted on either rectangular or triangular meshes, we can show the convergence accuracy is of order for the th order IMEX LDG scheme under consideration. Numerical experiments are also given to verify our main results.
Wang, Haijin 1 ; Wang, Shiping 2 ; Zhang, Qiang 1 ; Shu, Chi-Wang 3
@article{M2AN_2016__50_4_1083_0,
author = {Wang, Haijin and Wang, Shiping and Zhang, Qiang and Shu, Chi-Wang},
title = {Local discontinuous {Galerkin} methods with implicit-explicit time-marching for multi-dimensional convection-diffusion problems},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1083--1105},
publisher = {EDP-Sciences},
volume = {50},
number = {4},
year = {2016},
doi = {10.1051/m2an/2015068},
zbl = {1351.65078},
mrnumber = {3521713},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2015068/}
}
TY - JOUR AU - Wang, Haijin AU - Wang, Shiping AU - Zhang, Qiang AU - Shu, Chi-Wang TI - Local discontinuous Galerkin methods with implicit-explicit time-marching for multi-dimensional convection-diffusion problems JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 1083 EP - 1105 VL - 50 IS - 4 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2015068/ DO - 10.1051/m2an/2015068 LA - en ID - M2AN_2016__50_4_1083_0 ER -
%0 Journal Article %A Wang, Haijin %A Wang, Shiping %A Zhang, Qiang %A Shu, Chi-Wang %T Local discontinuous Galerkin methods with implicit-explicit time-marching for multi-dimensional convection-diffusion problems %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 1083-1105 %V 50 %N 4 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2015068/ %R 10.1051/m2an/2015068 %G en %F M2AN_2016__50_4_1083_0
Wang, Haijin; Wang, Shiping; Zhang, Qiang; Shu, Chi-Wang. Local discontinuous Galerkin methods with implicit-explicit time-marching for multi-dimensional convection-diffusion problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 4, pp. 1083-1105. doi: 10.1051/m2an/2015068
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