Local discontinuous Galerkin methods with implicit-explicit time-marching for multi-dimensional convection-diffusion problems
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 50 (2016) no. 4, pp. 1083-1105.

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 τ is only required to be upper-bounded by a positive constant independent of the spatial mesh size h. 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 k is adopted on either rectangular or triangular meshes, we can show the convergence accuracy is of order 𝒪 ( h k + 1 + τ s ) for the sth order IMEX LDG scheme ( s = 1 , 2 , 3 ) under consideration. Numerical experiments are also given to verify our main results.

DOI: 10.1051/m2an/2015068
Classification: 65M12, 65M15, 65M60
Keywords: Local discontinuous Galerkin method, implicit-explicit scheme, convection-diffusion, stability, error estimate
Wang, Haijin 1; Wang, Shiping 2; Zhang, Qiang 1; Shu, Chi-Wang 3

1 Department of Mathematics, Nanjing University, Jiangsu Province, Nanjing 210093, P.R. China.
2 College of Shipbuilding Engineering, Harbin Engineering University, Harbin 15000, P. R. China.
3 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA.
@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://www.numdam.org/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://www.numdam.org/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://www.numdam.org/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 , Volume 50 (2016) no. 4, pp. 1083-1105. doi : 10.1051/m2an/2015068. http://www.numdam.org/articles/10.1051/m2an/2015068/

U.M. Ascher, S.J. Ruuth and R.J. Spiteri, Implicit-explicit Runge−Kutta methods for time-dependent partial differential equations. Appl. Numer. Math. 25 (1997) 151–167. | DOI | MR | Zbl

F. Bassi and S. Rebay, A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier–Stokes equations, J. Comput. Phys. 131 (1997) 267–279. | DOI | MR | Zbl

M.P. Calvo, J. De Frutos and J. Novo, Linearly implicit Runge−Kutta methods for advection-reaction-diffusion equations. Appl. Numer. Math. 37 (2001) 535–549. | DOI | MR | Zbl

P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, New York (1978). | MR | Zbl

B. Cockburn and B. Dong, An analysis of the minimal dissipation local discontinuous Galerkin method for convection-diffusion problems, J. Sci. Comput. 32 (2007) 233–262. | DOI | MR | Zbl

B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35 (1998) 2440–2463. | DOI | MR | Zbl

B. Cockburn and C.-W. Shu, Runge−Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16 (2001) 173–261. | DOI | MR | Zbl

B. Cockburn, G. Kanschat, I. Perugia and D. Schötzau, Superconvergence of the local discontinuous Galerkin method for elliptic problems on rectangular grids. SIAM J. Numer. Anal. 39 (2001) 264–285. | DOI | MR | Zbl

B. Dong and C.-W. Shu, Analysis of a local discontinuous Galerkin methods for linear time-dependent fourth-order problems. SIAM J. Numer. Anal. 47 (2009) 3240–3268. | DOI | MR | Zbl

R.S. Falk and G.R. Richter, Analysis of a continuous finite element method for hyperbolic equations. SIAM J. Numer. Anal. 24 (1987) 257–278. | DOI | MR | Zbl

Y. Liu and C.-W. Shu, Analysis of the local discontinuous Galerkin method for the drift-diffusion model of semiconductor devices. Sci. China Math. 59 (2016) 115–140. | DOI | MR | Zbl

C.-W. Shu, Discontinuous Galerkin methods: general approach and stability, Numerical Solutions of Partial Differential Equations, Advanced Courses in Mathematics CRM Barcelona, edited by S. Bertoluzza, S. Falletta, G. Russo and C.-W. Shu. Birkhauser, Basel (2009) 149–201. | MR | Zbl

V. Thomḿe, Galerkin finite element methods for parabolic problems, 2nd edition. Springer Ser. Comput. Math. Springer-Verlag, Berlin (2007). | MR | Zbl

H.J. Wang and Q. Zhang, Error estimate on a fully discrete local discontinuous Galerkin method for linear convection-diffusion problem. J. Comput. Math. 31 (2013) 283–307. | DOI | MR | Zbl

H.J. Wang, C.-W. Shu and Q. Zhang, Stability and error estimates of the local discontinuous Galerkin method with implicit-explicit time-marching for advection-diffusion problems. SIAM J. Numer. Anal. 53 (2015) 206–227. | DOI | MR | Zbl

H.J. Wang, C.-W. Shu and Q. Zhang, Stability and error estimates of the local discontinuous Galerkin method with implicit-explicit time-marching for nonlinear convection-diffusion problems. Appl. Math. Comput. 272 (2015) 237–258. | MR

M.F. Wheeler, A priori L 2 error estimates for Galerkin approximations to parabolic partial differential equations. SIAM J. Numer. Anal. 10 (1973) 723–759. | DOI | MR | Zbl

Y.H. Xia, Y. Xu and C.-W. Shu, Efficient time discretization for local discontinuous Galerkin methods. Discrete Contin. Dyn. Syst. Ser. B 8 (2007) 677–693. | MR | Zbl

Y.H. Xia, Y. Xu and C.-W. Shu, Application of the local discontinuous Galerkin method for the Allen-Cahn/Cahn–Hilliard system. Commun. Comput. Phys. 5 (2009) 821–835. | MR | Zbl

Y. Xu and C.-W. Shu, Local discontinuous Galerkin methods for the Kuramoto–Sivashinsky equations and the Ito-type coupled KdV equations. Comput. Methods Appl. Mech. Engrg. 195 (2006) 3430–3447. | DOI | MR | Zbl

Y. Xu and C.-W. Shu, Local discontinuous Galerkin methods for high-order time-dependent partial differential equations. Commun. Comput. Phys. 7 (2010) 1–46. | MR | Zbl

J. Yan and C.-W. Shu, A local discontinuous Galerkin method for KdV type equations. SIAM J. Numer. Anal. 40 (2002) 769–791. | DOI | MR | Zbl

J. Yan and C.-W. Shu, Local discontinuous Galerkin methods for partial differential equations with higher order derivatives, J. Sci. Comput. 17 (2002) 17–27. | MR | Zbl

Q. Zhang and C.-W. Shu, Error estimates to smooth solution of Runge−Kutta discontinuous Galerkin methods for scalar conservation laws. SIAM J. Numer. Anal. 42 (2004) 641–666. | DOI | MR | Zbl

Q. Zhang and C.-W. Shu, Stability analysis and a priori error estimates to the third order explicit Runge−Kutta discontinuous Galerkin method for scalar conservation laws. SIAM J. Numer. Anal. 48 (2010) 1038–1063. | DOI | MR | Zbl

Cited by Sources: