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.

我们开发，分析和数值验证了局部不连续伽勒金（LDG）方法，用于求解非线性本杰明-波纳-马洪尼（BBM）方程。用适当地选择数值通量，可以示出传统的方法LDG保存质量的离散形式，并且或者保留或耗散能量的离散形式，高达四舍五入水平。通过一种发现辅助变量和主变量误差之间的联系的新技术，仔细研究了非线性BBM方程的半离散能量守恒和耗能方法，提供了具有最佳收敛阶数的误差估计。分析非线性项。完全离散的方法可以通过节能隐式中点时间离散化获得。数值实验证实了最佳收敛速度，以及质量和能量守恒/耗散特性。还比较了节能方法和耗能方法的长时间行为，以表明节能方法可以更好地逼近精确解。在Fu和Shu的最新研究（J Comput Phys 394：329–363，2019）中，针对线性对称双曲系统，开发了基于未知数加倍技术的最优节能不连续伽勒金方法。我们扩展了这一思想，为非线性BBM方程构造了另一类节能LDG方法。研究了它们的节能特性和最佳收敛速度（通过特殊构造的数值投影）。