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Optimal Energy Conserving and Energy Dissipative Local Discontinuous Galerkin Methods for the Benjamin–Bona–Mahony Equation

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Abstract

We develop, analyze and numerically validate local discontinuous Galerkin (LDG) methods for solving the nonlinear Benjamin–Bona–Mahony (BBM) equation. With appropriately chosen numerical fluxes, the conventional LDG methods can be shown to preserve the discrete version of mass, and either preserve or dissipate the discrete version of energy, up to the round-off level. The error estimate with optimal order of convergence is provided for both the semi-discrete energy conserving and energy dissipative methods applied to the nonlinear BBM equation, by a novel technique to discover the connection between the error of the auxiliary and primary variables, and by carefully analyzing the nonlinear term. Fully discrete methods can be derived with energy-conserving implicit midpoint temporal discretization. Numerical experiments confirm the optimal rates of convergence, as well as the mass and energy conserving/dissipative property. The comparison of the long time behavior of the energy conserving and energy dissipative methods are also provided, to show that the energy conserving method produces a better approximation to the exact solution. In a recent study by Fu and Shu (J Comput Phys 394:329–363, 2019), optimal energy conserving discontinuous Galerkin methods based on doubling-the-unknowns technique were developed for the linear symmetric hyperbolic systems. We extend the idea to construct another class of energy conserving LDG methods for the nonlinear BBM equation. Their energy conservation property and optimal convergence rate (via a special constructed numerical projection) are investigated. We also provide a comparison of these two types of energy conserving LDG methods, and shown that, under the same setup of computational elements, the latter method produces a smaller numerical error with slightly longer computational time.

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References

  1. Bona, J.L., Smith, R.: The initial-value problem for the Korteweg–de Vries equation. Philos. Trans. R. Soc. Lond. Ser. B Biol. Sci. 278, 555–601 (1975)

    MathSciNet  MATH  Google Scholar 

  2. Benjamin, T., Bona, J., Mahony, J.: Model equations for long waves in nonlinear dispersive systems. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Sci. 272, 47–78 (1972)

    MathSciNet  MATH  Google Scholar 

  3. Abbasbandy, S., Shirzadi, A.: The first integral method for modified Benjamin–Bona–Mahony equation. Commun. Nonlinear Sci. Numer. Simul. 15, 1759–1764 (2010)

    Article  MathSciNet  Google Scholar 

  4. Wazwaz, A.M.: Exact solution with compact and non-compact structures for the one-dimensional generalized Benjamin–Bona–Mahony equation. Commun. Nonlinear Sci. Numer. Simul. 10, 855–867 (2005)

    Article  MathSciNet  Google Scholar 

  5. Omrani, K.: The convergence of fully discrete Galerkin approximations for the Benjamin–Bona–Mahony (BBM) equation. Appl. Math. Comput. 180, 614–621 (2006)

    MathSciNet  MATH  Google Scholar 

  6. Abramowitz, M., Stegun, I.: editors, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, volume 55 of Applied Mathematics Series, National Bureau of Standards, MR167642 (1965)

  7. Avrin, J., Goldstein, J.A.: Global existence for the Benjamin–Bona–Mahony equation in arbitrary dimensions. Nonlinear Anal. 9, 861–865 (1985)

    Article  MathSciNet  Google Scholar 

  8. Goldstein, J.A., Wichnoski, B.: On the Benjamin–Bona–Mahony equation in higher dimensions. Nonlinear Anal. 4, 665–675 (1980)

    Article  MathSciNet  Google Scholar 

  9. Medeiros, L.A., Menzela, G.P.: Existence and uniqueness for periodic solutions of the Benjamin–Bona–Mahony equation. SIAM J. Math. Anal. 8, 792–799 (1977)

    Article  MathSciNet  Google Scholar 

  10. Wang, L., Zhou, J., Ren, L.: The exact solitary wave solutions for a family of BBM equation. Int. J. Nonlinear Sci. 1, 58–64 (2006)

    MathSciNet  MATH  Google Scholar 

  11. Achouri, T., Khiari, N., Omrani, K.: On the convergence of difference schemes for the Benjamin–Bona–Mahony (BBM) equation. Appl. Math. Comput. 182, 999–1005 (2006)

    MathSciNet  MATH  Google Scholar 

  12. Omrani, K., Ayadi, M.: Finite difference discretization of the Benjamin–Bona–Mahony–Burgers (BBMB) equation. Numer. Methods Partial Differ. Equ. 24, 239–248 (2008)

    Article  Google Scholar 

  13. Gao, F., Qiu, J., Zhang, Q.: Local discontinuous Galerkin finite element method and error estimates for one class of Sobolev equation. J. Sci. Comput. 41, 436–460 (2009)

    Article  MathSciNet  Google Scholar 

  14. Zhang, Q., Gao, F.: A fully-discrete local discontinuous Galerkin method for convection-dominated Sobolev equation. J. Sci. Comput. 51, 107–134 (2012)

    Article  MathSciNet  Google Scholar 

  15. Kamga, A., Mitsotakis, D.: A high order discontinuous Galerkin method for the convection dominated KdV–BBM equation and its efficient implementation, preprint available on researchgate.net

  16. Buli, J., Xing, Y.: Local discontinuous Galerkin methods for the Boussinesq coupled BBM system. J. Sci. Comput. 75, 536–559 (2018)

    Article  MathSciNet  Google Scholar 

  17. Cockburn, B., Hou, S., Shu, C.-W.: The Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case. Math. Comput. 54, 545–581 (1990)

    MathSciNet  MATH  Google Scholar 

  18. Cockburn, B., Karniadakis, G., Shu, C.-W.: The development of discontinuous Galerkin methods, In: Cockburn, B., Karniadakis, G., Shu, C.-W. (eds.) Discontinuous Galerkin Methods: Theory, Computation and Applications, pp. 3–50. Lecture Notes in Computational Science and Engineering, Part I: Overview, Vol. 11, Springer, Berlin (2000)

  19. Cockburn, B., Lin, S.-Y., Shu, C.-W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one dimensional systems. J. Comput. Phys. 84, 90–113 (1989)

    Article  MathSciNet  Google Scholar 

  20. 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)

    MathSciNet  MATH  Google Scholar 

  21. Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation, Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory (1973)

  22. Cockburn, B., Shu, C.W.: The local discontinuous Galerkin finite element method for convection-diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998)

    Article  MathSciNet  Google Scholar 

  23. Xu, Y., Shu, C.-W.: Local discontinuous Galerkin methods for high-order time-dependent partial differential equations. Commun. Comput. Phys. 7, 1–46 (2010)

    MathSciNet  MATH  Google Scholar 

  24. Bona, J.L., Chen, H., Karakashian, O.A., Xing, Y.: Conservative discontinuous Galerkin methods for the generalized Korteweg–de Vries equation. Math. Comput. 82, 1401–1432 (2013)

    Article  MathSciNet  Google Scholar 

  25. Yi, N., Huang, Y., Liu, H.: A direct discontinuous Galerkin method for the generalized Korteweg–de Vries equation: energy conservation and boundary effect. J. Comput. Phys. 242, 351–366 (2013)

    Article  MathSciNet  Google Scholar 

  26. Karakashian, O., Xing, Y.: A posteriori error estimates for conservative local discontinuous Galerkin methods for the generalized Korteweg–de Vries equation. Commun. Comput. Phys. 20, 250–278 (2016)

    Article  MathSciNet  Google Scholar 

  27. Zhang, Q., Xia, Y.: Conservative and dissipative local discontinuous Galerkin methods for Korteweg–de Vries type equations. Commun. Comput. Phys. 25, 532–563 (2019)

    MathSciNet  Google Scholar 

  28. Xing, Y., Chou, C.-S., Shu, C.-W.: Energy conserving local discontinuous Galerkin methods for wave propagation problems. Inverse Probl. Imaging 7, 967–986 (2013)

    Article  MathSciNet  Google Scholar 

  29. Chou, C.-S., Shu, C.-W., Xing, Y.: Optimal energy conserving local discontinuous Galerkin methods for second-order wave equation in heterogeneous media. J. Comput. Phys. 272, 88–107 (2014)

    Article  MathSciNet  Google Scholar 

  30. Cheng, Y., Chou, C.-S., Li, F., Xing, Y.: L2 stable discontinuous Galerkin methods for one-dimensional two-way wave equations. Math. Comput. 86, 121–155 (2017)

    Article  Google Scholar 

  31. Liu, H., Xing, Y.: An invariant preserving discontinuous Galerkin method for the Camassa–Holm equation. SIAM J. Sci. Comput. 38, A1919–A1934 (2016)

    Article  MathSciNet  Google Scholar 

  32. Huang, Y., Liu, H., Yi, N.: A conservative discontinuous Galerkin method for the Degasperis-Procesi equation. Methods Appl. Anal. 21, 67–90 (2014)

    MathSciNet  MATH  Google Scholar 

  33. Guo, L., Xu, Y.: Energy conserving local discontinuous Galerkin methods for the nonlinear Schrödinger equation with wave operator. J. Sci. Comput. 65, 622–647 (2015)

    Article  MathSciNet  Google Scholar 

  34. Liang, X., Khaliq, A.Q.M., Xing, Y.: Fourth order exponential time differencing method with local discontinuous Galerkin approximation for coupled nonlinear Schrödinger equations. Commun. Comput. Phys. 17, 510–541 (2015)

    Article  MathSciNet  Google Scholar 

  35. Li, X., Sun, W., Xing, Y., Chou, C.-S.: Energy conserving local discontinuous Galerkin methods for the improved Boussinesq equation. J. Comput. Phys. 401, 109002 (2020)

    Article  MathSciNet  Google Scholar 

  36. Fu, G., Shu, C.-W.: Optimal energy-conserving discontinuous Galerkin methods for linear symmetric hyperbolic systems. J. Comput. Phys. 394, 329–363 (2019)

    Article  MathSciNet  Google Scholar 

  37. Fu, G., Shu, C.-W.: An energy-conserving ultra-weak discontinuous Galerkin method for the generalized Korteweg–de Vries equation. J. Comput. Appl. Math. 349, 41–51 (2019)

    Article  MathSciNet  Google Scholar 

  38. 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, 1225–1261 (2016)

    Article  MathSciNet  Google Scholar 

  39. Cheng, Y., Meng, X., Zhang, Q.: Application of generalized Gauss–Radau projections for the local discontinuous Galerkin method for linear convection-diffusion equations. Math. Comput. 86, 1233–1267 (2017)

    Article  MathSciNet  Google Scholar 

  40. Xu, Y., Shu, C.-W.: Error estimates of the semi-discrete local discontinuous Galerkin method for nonlinear convection diffusion and KdV equations. Comput. Methods Appl. Mech. Eng. 196, 3805–3822 (2007)

    Article  MathSciNet  Google Scholar 

  41. Wang, H., Shu, C.-W., Zhang, Q.: Stability and error estimates of local discontinuous Galerkin method with implicit-explicit time-marching for advection-diffusion problems. SIAM J. Numer. Anal. 53, 206–227 (2015)

    Article  MathSciNet  Google Scholar 

  42. Li, X.H., Shu, C.-W., Yang, Y.: Local discontinuous Galerkin method for the Keller–Segel chemotaxis model. J. Sci. Comput. 73, 943–967 (2017)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. College of Automation, Harbin Engineering University, Harbin, 150001, China

    Xiaole Li

  2. Department of Mathematics, The Ohio State University, Columbus, OH, 43210, USA

    Xiaole Li, Yulong Xing & Ching-Shan Chou

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  1. Xiaole Li
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  2. Yulong Xing
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Correspondence to Yulong Xing.

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XL is supported by the College of Automation at Harbin Engineering University, the National Natural Science Foundation of China (No. 11801116) and the Fundamental Research Funds for the Central Universities. YX is partially supported by the NSF Grant DMS-1753581. CSC is supported by the NSF Grants DMS-1253481 and DMS-1813071.

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Li, X., Xing, Y. & Chou, CS. Optimal Energy Conserving and Energy Dissipative Local Discontinuous Galerkin Methods for the Benjamin–Bona–Mahony Equation. J Sci Comput 83, 17 (2020). https://doi.org/10.1007/s10915-020-01172-6

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  • DOI: https://doi.org/10.1007/s10915-020-01172-6

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