Journal of Scientific Computing

, Volume 78, Issue 3, pp 1660–1690 | Cite as

Local Discontinuous Galerkin Methods with Generalized Alternating Numerical Fluxes for Two-dimensional Linear Sobolev Equation

  • Di Zhao
  • Qiang ZhangEmail author


In this paper we present an efficient and high-order numerical method to solve two-dimensional linear Sobolev equations, which is based on the local discontinuous Galerkin (LDG) method with the upwind-biased fluxes and generalized alternating fluxes. A weak stability is given for both schemes, and a strong stability is established if the initial solutions exactly satisfy the elemental discontinuous Galerkin discretization. Moreover, the sharp error estimate in \(L^2\)-norm is established, by an elaborate application of the generalized Gauss–Radau projection. A fully-discrete LDG scheme is also considered, where the third-order explicit TVD Runge–Kutta algorithm is adopted. Finally some numerical experiments are given.


Sobolev equation Local discontinuous Galerkin method Upwind-biased/generalized alternating numerical fluxes Stability and error estimate Generalized Gauss–Radau projection 


  1. 1.
    Amiraliyev, G.M., Mamedov, Y.D.: Difference schemes on the uniform mesh for singular perturbed pseudo-parabolic equations. Turk. J. Math. 19, 207–222 (1995)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Arnold, D.N., Douglas, J., Thomee, V.: Superconvergence of finite element approximation to the solution of a Sobolev equation in a single space variable. Math. Comput. 27, 737–743 (1981)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bassi, F., Rebay, S.: A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier–Stokes equations. J. Comput. Phys. 131, 267–279 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cheng, Y., Meng, X., Zhang, Q.: Application of generalized Gauss–Radau projections for the local discontinuous Galerkin methods for linear convection–diffusion equations. Math. Comput. 86, 1233–1267 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cockburn, B., Shu, C.W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math. Comput. 52, 411–435 (1989)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Cockburn, B., Shu, C.W.: The Runge–Kutta discontinuous Galerkin finite element method for conservation laws V: multidimensional systems. J. Comput. Phys. 141, 199–2224 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cockburn, B., Shu, C.W.: The local discontinuous Galerkin method for time-dependent convection–diffusion systems. SIAM. J. Numer. Anal. 35, 2440–2463 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cockburn, B., Shu, C.W.: Runge–Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16, 173–261 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Davis, P.L.: A quasilinear parabolic and a related third order problem. J. Math. Anal. Appl. 40, 327–335 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Esen, A., Kutluay, S.: Application of a lumped Galerkin method to the regularized long wave equation. Appl. Math. Comput. 174, 485–494 (2001)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Ewing, R.E.: Numerical solution of Sobolev partial differential equations. SIAM J. Numer. Anal. 12, 345–363 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ewing, R.E.: A coupled non-linear hyperbolic-Sobolev system. Ann. Mat. Pure Appl. 114, 331–349 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ewing, R.E.: Time-stepping Galerkin methods for nonlinear Sobolev partial differential equation. SIAM. J. Numer. Anal. 15, 1125–1150 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gao, F.Z., Qiu, J.X., Zhang, Q.: Local discontinuous Galerkin finite element method and error estimates for one class of Sobolev equation. J. Sci. Comput. 41, 436–460 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Guo, L., Chen, H.: \(H_{1}\)-Galerkin mixed finite element method for the regularized long wave equation. Computing 77, 205–221 (2006)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Levy, D., Shu, C.W., Yan, J.: Local discontinuous Galerkin methods for nonlinear dispersive equations. J. Comput. Phys. 196, 751–774 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Meng, X., Shu, C.W., Wu, B.: Optimal error estimates for discontinuous Galerkin methods based on upwind-biased fluxes for linear hyperbolic equations. Math. Comput. 85, 1225–1661 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Omrani, K.: The convergence of the fully discrete Galerkin approximations for Benjamin–Bona–Mahony equation. Appl. Math. Comput. 180, 614–621 (2006)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Pani, A.K., Fairweather, G.: \(H_{1}\)-Galerkin mixed finite element methods for parabolic partial integro-differential equations. IMAJ Numer. Anal. 22, 231–252 (2002)CrossRefzbMATHGoogle Scholar
  20. 20.
    Xu, Y., Shu, C.W.: A local discontinuous Galerkin method for the Camassa–Holm equation. SIAM. J. Numer. Anal. 46(4), 1998–2021 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Yan, J., Shu, C.W.: Local discontinuous Galerkin method for partial differential equations with higher order derivatives. J. Sci. Comput. 17, 27–47 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Yan, J., Shu, C.W.: A local discontinuous Galerkin method for KdV type equations. SIAM. J. Numer. Anal. 40, 769–791 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Zhang, Q., Gao, F.Z.: A fully-discrete local discontinuous Galerkin method for convection-dominated Sobolev equation. J. Sci. Comput. 51(1), 107–134 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Zhang, Q., Shu, C.W.: Stability analysis and a priori error estimates of third order explicit Runge–Kutta discontinuous Galerkin method for scalar conservation laws. SIAM. J. Numer. Anal. 48, 1038–1063 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsNanjing UniversityNanjingPeople’s Republic of China

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