Optimal Energy Conserving and Energy Dissipative Local Discontinuous Galerkin Methods for the Benjamin–Bona–Mahony Equation
Optimal Energy Conserving and Energy Dissipative Local Discontinuous Galerkin Methods for the Benjamin–Bona–Mahony Equation
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DOI：
10.1007/s10915-020-01172-6
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发表时间：
2020-04
影响因子：
2.5
通讯作者：
Xiaole Li;Y. Xing;Ching-Shan Chou
中科院分区：
文献类型：
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作者：
Xiaole Li;Y. Xing;Ching-Shan Chou
我们开发，分析和数值验证局部间断Galerkin（LDG）方法求解非线性Benjamin-Bona-Mahony（BBM）方程。适当选择的数值通量，传统的LDG方法可以被证明，以保持离散版本的质量，并保持或耗散离散版本的能量，直到四舍五入的水平。通过一种新的技术来发现辅助变量和主变量误差之间的联系，并通过仔细分析非线性项，为应用于非线性BBM方程的半离散能量守恒和能量耗散方法提供了具有最佳收敛阶的误差估计.全离散方法可以通过能量守恒的隐式中点时间离散化得到。数值实验证实了最佳的收敛速度，以及质量和能量守恒/耗散属性。通过对能量守恒法和能量耗散法的长时间行为的比较，说明能量守恒法能更好地逼近精确解。在Fu和Shu最近的一项研究中（J Comput Phys 394：329-363，2019），基于未知数加倍技术的最优能量守恒间断Galerkin方法被开发用于线性对称双曲型系统。我们将这一思想推广到非线性BBM方程，构造了另一类能量守恒的LDG方法。研究了它们的能量守恒性质和最优收敛速度（通过一个特殊构造的数值投影）。我们还提供了这两种类型的节能LDG方法的比较，并表明，在相同的设置的计算元素，后一种方法产生较小的数值误差与稍长的计算时间。
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.