Stability and Error Estimates of a Novel Spectral Deferred Correction Time-Marching with Local Discontinuous Galerkin Methods for Parabolic Equations

References

[1] 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. 10.1016/S0168-9274(97)00056-1Search in Google Scholar

[2] 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), no. 2, 267–279. 10.1006/jcph.1996.5572Search in Google Scholar

[3] M. P. Calvo, J. de Frutos and J. Novo, Linearly implicit Runge-Kutta methods for advection-reaction-diffusion equations, Appl. Numer. Math. 37 (2001), no. 4, 535–549. 10.1016/S0168-9274(00)00061-1Search in Google Scholar

[4] P. Castillo, An optimal estimate for the local discontinuous Galerkin method, Discontinuous Galerkin Methods, Lect. Notes Comput. Sci. Eng. 11, Springer, Berlin (2000), 285–290. 10.1007/978-3-642-59721-3_23Search in Google Scholar

[5] P. Castillo, B. Cockburn, D. Schötzau and C. Schwab, Optimal a priori error estimates for the -version of the local discontinuous Galerkin method for convection-diffusion problems, Math. Comp. 71 (2002), no. 238, 455–478. 10.1090/S0025-5718-01-01317-5Search in Google Scholar

[6] Y. Cheng, X. Meng and Q. Zhang, Application of generalized Gauss–Radau projections for the local discontinuous Galerkin method for linear convection-diffusion equations, Math. Comp. 86 (2017), no. 305, 1233–1267. 10.1090/mcom/3141Search in Google Scholar

[7] Y. Cheng and Q. Zhang, Local analysis of the local discontinuous Galerkin method with generalized alternating numerical flux for one-dimensional singularly perturbed problem, J. Sci. Comput. 72 (2017), no. 2, 792–819. 10.1007/s10915-017-0378-ySearch in Google Scholar

[8] B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal. 35 (1998), no. 6, 2440–2463. 10.1137/S0036142997316712Search in Google Scholar

[9] B. Dong and C.-W. Shu, Analysis of a local discontinuous Galerkin method for linear time-dependent fourth-order problems, SIAM J. Numer. Anal. 47 (2009), no. 5, 3240–3268. 10.1137/080737472Search in Google Scholar

[10] P. Fu, Y. Cheng, F. Li and Y. Xu, Discontinuous Galerkin methods with optimal accuracy for one dimensional linear PDEs with high order spatial derivatives, J. Sci. Comput. 78 (2019), no. 2, 816–863. 10.1007/s10915-018-0788-5Search in Google Scholar

[11] 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), no. 3, 1029–1054. 10.1007/s10915-016-0170-4Search in Google Scholar

[12] R. Guo, Y. Xia and Y. Xu, Semi-implicit spectral deferred correction methods for highly nonlinear partial differential equations, J. Comput. Phys. 338 (2017), 269–284. 10.1016/j.jcp.2017.02.059Search in Google Scholar

[13] R. Guo and Y. Xu, Local discontinuous Galerkin method and high order semi-implicit scheme for the phase field crystal equation, SIAM J. Sci. Comput. 38 (2016), no. 1, A105–A127. 10.1137/15M1038803Search in Google Scholar

[14] R. Guo and Y. Xu, Semi-implicit spectral deferred correction methods based on second order time integration schemes for nonlinear PDEs, J. Comput. Math., to appear. Search in Google Scholar

[15] D. Levy, C.-W. Shu and J. Yan, Local discontinuous Galerkin methods for nonlinear dispersive equations, J. Comput. Phys. 196 (2004), no. 2, 751–772. 10.1016/j.jcp.2003.11.013Search in Google Scholar

[16] X. Meng, C.-W. Shu and B. Wu, Optimal error estimates for discontinuous Galerkin methods based on upwind-biased fluxes for linear hyperbolic equations, Math. Comp. 85 (2016), no. 299, 1225–1261. 10.1090/mcom/3022Search in Google Scholar

[17] C.-W. Shu, Discontinuous Galerkin methods: General approach and stability, Numerical Solutions of Partial Differential Equations, Adv. Courses Math. CRM Barcelona, Birkhäuser, Basel (2009), 149–201. Search in Google Scholar

[18] H. 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), no. 1, 206–227. 10.1137/140956750Search in Google Scholar

[19] H. 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. 10.1016/j.amc.2015.02.067Search in Google Scholar

[20] H. Wang, Q. Zhang and C.-W. Shu, Stability analysis and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for the time-dependent fourth order PDEs, ESAIM Math. Model. Numer. Anal. 51 (2017), no. 5, 1931–1955. 10.1051/m2an/2017017Search in Google Scholar

[21] H. Wang, Q. Zhang and C.-W. Shu, Implicit-explicit local discontinuous Galerkin methods with generalized alternating numerical fluxes for convection-diffusion problems, J. Sci. Comput. 81 (2019), no. 3, 2080–2114. 10.1007/s10915-019-01072-4Search in Google Scholar

[22] Y. Xia, Y. Xu and C.-W. Shu, Efficient time discretization for local discontinuous Galerkin methods, Discrete Contin. Dyn. Syst. Ser. B 8 (2007), no. 3, 677–693. 10.3934/dcdsb.2007.8.677Search in Google Scholar

[23] Y. Xu and C.-W. Shu, Local discontinuous Galerkin methods for three classes of nonlinear wave equations, J. Comput. Math. 22 (2004), 250–274. Search in Google Scholar

[24] Y. Xu and C.-W. Shu, Local discontinuous Galerkin methods for high-order time-dependent partial differential equations, Commun. Comput. Phys. 7 (2010), no. 1, 1–46. Search in Google Scholar

[25] J. Yan and C.-W. Shu, A local discontinuous Galerkin method for KdV type equations, SIAM J. Numer. Anal. 40 (2002), no. 2, 769–791. 10.1137/S0036142901390378Search in Google Scholar

[26] L. Zhou and Y. Xu, Stability analysis and error estimates of semi-implicit spectral deferred correction coupled with local discontinuous Galerkin method for linear convection-diffusion equations, J. Sci. Comput. 77 (2018), no. 2, 1001–1029. 10.1007/s10915-018-0735-5Search in Google Scholar