Stability analysis and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for the time-dependent fourth order PDEs

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

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 5, pp. 1931-1955.

The main purpose of this paper is to give stability analysis and error estimates of the local discontinuous Galerkin (LDG) methods coupled with three specific implicit-explicit (IMEX) Runge–Kutta time discretization methods up to third order accuracy, for solving one-dimensional time-dependent linear fourth order partial differential equations. In the time discretization, all the lower order derivative terms are treated explicitly and the fourth order derivative term is treated implicitly. By the aid of energy analysis, we show that the IMEX-LDG schemes are unconditionally energy stable, in the sense that the time step $\tau$ is only required to be upper-bounded by a constant which is independent of the mesh size $h$. The optimal error estimate is also derived by the aid of the elliptic projection and the adjoint argument. Numerical experiments are given to verify that the corresponding IMEX-LDG schemes can achieve optimal error accuracy.

Classification : 65M12, 65M15, 65M60
Mots clés : Local discontinuous Galerkin method, implicit-explicit time-marching scheme, time-dependent fourth order equations, stability, error estimates

R.A. Adams, Sobolev Spaces. New York, Academic Press (1975). | Zbl

D.N. Arnold, An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19 (1982) 742–760. | DOI | MR | Zbl

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 | 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 | Zbl

H. Blum and R. Rannacher, On the boundary value problem of the biharmonic operator on domains with angular corners. Math. Methods Appl. Sci. 2 (1980) 556–581. | DOI | Zbl

P. Castillo, B. Cockburn, I. Perugia and D. Schötzau, An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal. 38 (2000) 1676–1706. | DOI | Zbl

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

R. Guo, F. Filbet and Y. Xu, Efficient high order semi-implicit time discretization and local discontinuous Galerkin methods for highly nonlinear PDEs. J. Sci. Comput. 68 (2016) 1–26. | Zbl

J. Kadlec, On the regularity of the solution of the Poisson problem on a domain with boundary locally similar to the boundary of a convex set. Czech. Math. J. 14 (1964) 386–393. | DOI | Zbl

D. Levy, C.-W. Shu and J. Yan, Local discontinuous Galerkin methods for nonlinear dispersive equations. J. Comput. Phys. 196 (2004) 751–772. | DOI | Zbl

Y.X. 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 | 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 | Zbl

H.J. Wang, C.-W. Shu and Q. Zhang, Stability analysis and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for nonlinear convection-diffusion problems. Appl. Math. Comput. 272 (2016) 237–258. | Zbl

H.J. Wang, S.P. Wang, Q. Zhang and C.-W. Shu, Local discontinuous Galerkin methods with implicit-explicit time-marching for multi-dimensional convection-diffusion problems. ESAIM: M2AN 50 (2016) 1083–1105. | DOI | Numdam | 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. | Zbl

Y.H. Xia, Y. Xu and C.W. Shu, Local discontinuous Galerkin methods for the Cahn-Hilliard type equations. J. Comput. Phys. 227 (2007) 472–491. | DOI | Zbl

Y. Xu and C.-W. Shu, Local discontinuous Galerkin methods for three classes of nonlinear wave equations. J. Comput. Math. 22 (2004) 250–274. | 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. | 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 | 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 | Zbl

J. Yan and C.-W. Shu, Local discontinuous Galerkin methods for partial differential equations with higher order derivatives. J. Sci. Comput. 17 (2002) 27–47. | DOI | MR | Zbl

Q. Zhang and F. Z. Gao, A fully-discrete local discontinuous Galerkin method for convection-dominated Sobolev equation. J. Sci. Comput. 51 (2012) 107–134. | DOI | MR | Zbl