色散Wave Equations的一类守恒数值方法
A Broad Class of Conservative Numerical Methods for Dispersive Wave Equations
摘要：
我们开发了一个通用的框架，设计保守的数值方法的基础上求和的部分运营商和分裂形式在空间中，结合松弛龙格-库塔法在时间上。我们应用这个框架来创建几个非线性色散波方程的新类的全离散保守方法：Benjamin-Bona-Mahony（BBM），Fornberg-Whitham，Camassa-Holm，Degasperis-Procesi，Holm-Hone和BBM-BBM系统。这些完整的离散保存所有的线性不变量和一个非线性不变量的每个系统。空间半离散方法包括有限差分法、谱配置法、间断和连续有限元法。时间离散基本上是显式的，使用松弛龙格-库塔法。我们实现了一些具体的计划，从派生类，并通过数值试验证明其良好的性能。
英文摘要：
We develop a general framework for designing conservative numerical methods based on summation by parts operators and split forms in space, combined with relaxation Runge-Kutta methods in time. We apply this framework to create new classes of fully-discrete conservative methods for several nonlinear dispersive wave equations: Benjamin-Bona-Mahony (BBM), Fornberg-Whitham, Camassa-Holm, Degasperis-Procesi, Holm-Hone, and the BBM-BBM system. These full discretizations conserve all linear invariants and one nonlinear invariant for each system. The spatial semidiscretizations include finite difference, spectral collocation, and both discontinuous and continuous finite element methods. The time discretization is essentially explicit, using relaxation Runge-Kutta methods. We implement some specific schemes from among the derived classes, and demonstrate their favorable properties through numerical tests.
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