American Institute of Mathematical Sciences

Advanced
October  2007, 8(3): 677-693. doi: 10.3934/dcdsb.2007.8.677

Efficient time discretization for local discontinuous Galerkin methods

Yinhua Xia 1, , Yan Xu 1, and  Chi-Wang Shu 2,

1. 

Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China, China
2. 

Division of Applied Mathematics, Brown University, Providence, RI 02912, United States

Received  July 2006 Revised  January 2007 Published  July 2007

In this paper, we explore three efficient time discretization techniques for the local discontinuous Galerkin (LDG) methods to solve partial differential equations (PDEs) with higher order spatial derivatives. The main difficulty is the stiffness of the LDG spatial discretization operator, which would require a unreasonably small time step for an explicit local time stepping method. We focus our discussion on the semi-implicit spectral deferred correction (SDC) method, and study its stability and accuracy when coupled with the LDG spatial discretization. We also discuss two other time discretization techniques, namely the additive Runge-Kutta (ARK) method and the exponential time differencing (ETD) method, coupled with the LDG spatial discretization. A comparison is made among these three time discretization techniques, to conclude that all three methods are efficient when coupled with the LDG spatial discretization for solving PDEs containing higher order spatial derivatives. In particular, the SDC method has the advantage of easy implementation for arbitrary order of accuracy, and the ARK method has the smallest CPU cost in our implementation.
Keywords: local discontinuous Galerkin method, additive Runge-Kutta method, higher order spatial derivatives., Spectral deferred correction method, exponential time differencing method.
Mathematics Subject Classification: Primary: 65L06, 65M6.
Citation: Yinhua Xia, Yan Xu, Chi-Wang Shu. Efficient time discretization for local discontinuous Galerkin methods. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 677-693. doi: 10.3934/dcdsb.2007.8.677
[1]

Sihong Shao, Huazhong Tang. Higher-order accurate Runge-Kutta discontinuous Galerkin methods for a nonlinear Dirac model. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 623-640. doi: 10.3934/dcdsb.2006.6.623
[2]

Chaoxu Pei, Mark Sussman, M. Yousuff Hussaini. A space-time discontinuous Galerkin spectral element method for the Stefan problem. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3595-3622. doi: 10.3934/dcdsb.2017216
[3]

Yoshifumi Aimoto, Takayasu Matsuo, Yuto Miyatake. A local discontinuous Galerkin method based on variational structure. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 817-832. doi: 10.3934/dcdss.2015.8.817
[4]

Wenjuan Zhai, Bingzhen Chen. A fourth order implicit symmetric and symplectic exponentially fitted Runge-Kutta-Nyström method for solving oscillatory problems. Numerical Algebra, Control & Optimization, 2019, 9 (1) : 71-84. doi: 10.3934/naco.2019006
[5]

Antonia Katzouraki, Tania Stathaki. Intelligent traffic control on internet-like topologies - integration of graph principles to the classic Runge--Kutta method. Conference Publications, 2009, 2009 (Special) : 404-415. doi: 10.3934/proc.2009.2009.404
[6]

Mahboub Baccouch. Superconvergence of the semi-discrete local discontinuous Galerkin method for nonlinear KdV-type problems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 19-54. doi: 10.3934/dcdsb.2018104
[7]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 4907-4926. doi: 10.3934/dcdsb.2020319
[8]

Zheng Sun, José A. Carrillo, Chi-Wang Shu. An entropy stable high-order discontinuous Galerkin method for cross-diffusion gradient flow systems. Kinetic & Related Models, 2019, 12 (4) : 885-908. doi: 10.3934/krm.2019033
[9]

Na An, Chaobao Huang, Xijun Yu. Error analysis of discontinuous Galerkin method for the time fractional KdV equation with weak singularity solution. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 321-334. doi: 10.3934/dcdsb.2019185
[10]

Armando Majorana. A numerical model of the Boltzmann equation related to the discontinuous Galerkin method. Kinetic & Related Models, 2011, 4 (1) : 139-151. doi: 10.3934/krm.2011.4.139
[11]

Zexuan Liu, Zhiyuan Sun, Jerry Zhijian Yang. A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction. Electronic Research Archive, 2020, 28 (4) : 1487-1501. doi: 10.3934/era.2020078
[12]

Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, 2021, 29 (3) : 2375-2389. doi: 10.3934/era.2020120
[13]

Yanzhao Cao, Li Yin. Spectral Galerkin method for stochastic wave equations driven by space-time white noise. Communications on Pure & Applied Analysis, 2007, 6 (3) : 607-617. doi: 10.3934/cpaa.2007.6.607
[14]

Torsten Keßler, Sergej Rjasanow. Fully conservative spectral Galerkin–Petrov method for the inhomogeneous Boltzmann equation. Kinetic & Related Models, 2019, 12 (3) : 507-549. doi: 10.3934/krm.2019021
[15]

Yuhua Zhu. A local sensitivity and regularity analysis for the Vlasov-Poisson-Fokker-Planck system with multi-dimensional uncertainty and the spectral convergence of the stochastic Galerkin method. Networks & Heterogeneous Media, 2019, 14 (4) : 677-707. doi: 10.3934/nhm.2019027
[16]

Shi Jin, Yingda Li. Local sensitivity analysis and spectral convergence of the stochastic Galerkin method for discrete-velocity Boltzmann equations with multi-scales and random inputs. Kinetic & Related Models, 2019, 12 (5) : 969-993. doi: 10.3934/krm.2019037
[17]

Da Xu. Numerical solutions of viscoelastic bending wave equations with two term time kernels by Runge-Kutta convolution quadrature. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2389-2416. doi: 10.3934/dcdsb.2017122
[18]

Danijela Damjanovic and Anatole Katok. Local rigidity of actions of higher rank abelian groups and KAM method. Electronic Research Announcements, 2004, 10: 142-154.

[19]

ShinJa Jeong, Mi-Young Kim. Computational aspects of the multiscale discontinuous Galerkin method for convection-diffusion-reaction problems. Electronic Research Archive, 2021, 29 (2) : 1991-2006. doi: 10.3934/era.2020101
[20]

Kim S. Bey, Peter Z. Daffer, Hideaki Kaneko, Puntip Toghaw. Error analysis of the p-version discontinuous Galerkin method for heat transfer in built-up structures. Communications on Pure & Applied Analysis, 2007, 6 (3) : 719-740. doi: 10.3934/cpaa.2007.6.719

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (204)
  • HTML views (0)
  • Cited by (39)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy