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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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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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