Local discontinuous Galerkin methods with implicit-explicit time-marching for multi-dimensional convection-diffusion problems

Wang, Haijin; Wang, Shiping; Zhang, Qiang; Shu, Chi-Wang

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 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 $\tau$ 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 $𝒪(h^{k+1}+{\tau }^s)$ for the $s$th order IMEX LDG scheme $(s=1,2,3)$ under consideration. Numerical experiments are also given to verify our main results.

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