Error estimates of the third order runge-kutta alternating evolution discontinuous galerkin method for convection-diffusion problems

Liu, Hailiang¹ ; Wen, Hairui¹

¹

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 5, pp. 1709-1732.

Résumé

In this paper, we present the stability analysis and error estimates for the alternating evolution discontinuous Galerkin (AEDG) method with third order explicit Runge-Kutta temporal discretization for linear convection-diffusion equations. The scheme is shown stable under a CFL-like stability condition $c_0\tau \le ϵ\le c_1{h}^2$. Here $ϵ$ is the method parameter, and $h$ is the maximum spatial grid size. We further obtain the optimal $L^2$ error of order $O({\tau }^3+h^{k+1})$. Key tools include two approximation finite element spaces to distinguish overlapping polynomials, coupled global projections, and energy estimates of errors. For completeness, the stability analysis and error estimates for second order explicit Runge-Kutta temporal discretization is included in the appendix.

Détail

DOI : 10.1051/m2an/2018020

Classification : 65M15, 65M60, 35K20
Keywords: Alternating evolution, convection-diffusion equation, discontinuous Galerkin, error estimates, Runge-Kutta method

Bibliographie

[1] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, 3rd edn. Springer (2008). | Zbl | MR | DOI

[2] B. Cockburn and C.-W. Shu, Runge-Kutta dicontinuous Galerkin Methods for convection-dominated problems. J. Sci. Comput. 16 (2004) 173–261. | Zbl | MR | DOI

[3] S. Gottlieb, C.-W. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods. SIAM Rev. 43 (2001) 89–112. | Zbl | MR | DOI

[4] F. Li and S. Yakovlev, A central discontinuous Galerkin method for Hamilton-Jacobi equations. J. Sci. Comput. 45 (2010) 404–428. | Zbl | MR | DOI

[5] Y.-J. Liu, Central schemes on overlapping cells. J. Comput. Phys. 209 (2005) 82–104. | Zbl | MR | DOI

[6] H. Liu, An alternating evolution approximation to systems of hyperbolic conservation laws. J. Hyperbolic Differ. Equ. 5 (2008) 1–27. | Zbl | MR

[7] H. Liu and M. Pollack, Alternating evolution discontinuous Galerkin methods for Hamilton-Jacobi equations. J. Comput. Phys. 258 (2014) 31–46. | Zbl | MR | DOI

[8] H. Liu and M. Pollack, Alternating evolution discontinuous Galerkin methods for convection-diffusion equations. J. Comput. Phys. 307 (2016) 574–592. | Zbl | MR | DOI

[9] H. Liu and H.R. Wen, Error estimates for the AEDG method to one-dimensional linear convection-diffusion equations. Math. Comput. 87 (2018) 123–148. | Zbl | MR | DOI

[10] Y.-J. Liu, C.-W. Shu, E. Tadmor and M.-P. Zhang, Central discontinuous Galerkin methods on overlapping cells with a nonoscillatory hierarchical reconstruction. SIAM J. Numer. Anal. 45 (2007) 2442–2467. | Zbl | MR | DOI

[11] Y.-J. Liu,C.-W. Shu, E. Tadmor and M.-P. Zhang, L² stability analysis of the central discontinuous Galerkin method and a comparison between the central and regular discontinuous Galerkin methods. ESAIM: M2AN 42 (2008) 593–607 | Zbl | MR | mathdoc-id | DOI

[12] Y.-J. Liu, C.-W. Shu, E. Tadmor and M.-P. Zhang, Central local discontinuous Galerkin methods on overlapping cells for diffusion equations. ESAIM: M2AN 45 (2011) 1009–1032 | Zbl | MR | mathdoc-id | DOI

[13] H. Liu, M. Pollack and H. Saran, Alternating evolution discontinuous schemes for Hamilton-Jacobi equations. SIAM J. Sci. Comput. 35 (2013) 122–149. | Zbl | MR | DOI

[14] M.A. Reyna and F. Li, Operator bounds and time step conditions for the DG and central DG methods. J. Sci. Comput. 62 (2015) 532–534. | Zbl | MR | DOI

[15] H. Saran and H. Liu, Formulation and analysis of alternating evolution schemes for conservation laws. SIAM J. Sci. Comput. 33 (2011) 3210–40. | Zbl | MR | DOI

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

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

[18] Q. Zhang and C.-W. Shu, Error estimates to smooth solutions of Runge-Kutta discontinuous Galerkin methods for symmetrizable systems of conservation laws. SIAM J. Numer. Anal. 44 (2006) 1703–1720. | Zbl | MR | DOI

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

[20] Q. Zhang and C.-W. Shu, Error estimates to smooth solutions of Runge-Kutta discontinuous Galerkin methods for scalar conservation laws. SIAM J. Numer. Anal. 42 (2014) 641–666. | Zbl | MR | DOI