论文标题
一种适用于正规浅水波方程的保守的全差异数值方法
A conservative fully-discrete numerical method for the regularised shallow water wave equations
论文作者
论文摘要
本文提出了一种新的，保守的全差异方案，用于在周期性和反射性边界条件的情况下，正规化浅水浅水布斯西斯方程式的数值解决方案。该特定系统是最近得出的一类方程之一，可用于描述弱非线性和弱分散长水波（例如海啸）的传播。小振幅长波的研究通常需要长时间的模拟，以便研究场景，例如超过两个孤立波的碰撞或跨波动海啸的传播。对于长期模拟了非隔离波，例如孤立波，通过数值方法保留总能量对于近似质量至关重要。新的保守完全消散的方法包括用于空间半差异化的盖尔金有限元方法和及时整合的显式松弛runge-kutta方案。 Galerkin方法在混合有限元方法的框架中表达和实现。本文提供了对新数值方法的准确性和收敛性的扩展实验研究。实验揭示了与标准盖尔金方法相比的新收敛模式。
The paper proposes a new, conservative fully-discrete scheme for the numerical solution of the regularised shallow water Boussinesq system of equations in the cases of periodic and reflective boundary conditions. The particular system is one of a class of equations derived recently and can be used in practical simulations to describe the propagation of weakly nonlinear and weakly dispersive long water waves, such as tsunamis. Studies of small-amplitude long waves usually require long-time simulations in order to investigate scenarios such as the overtaking collision of two solitary waves or the propagation of transoceanic tsunamis. For long-time simulations of non-dissipative waves such as solitary waves, the preservation of the total energy by the numerical method can be crucial in the quality of the approximation. The new conservative fully-discrete method consists of a Galerkin finite element method for spatial semidiscretisation and an explicit relaxation Runge--Kutta scheme for integration in time. The Galerkin method is expressed and implemented in the framework of mixed finite element methods. The paper provides an extended experimental study of the accuracy and convergence properties of the new numerical method. The experiments reveal a new convergence pattern compared to standard Galerkin methods.
