This paper generalizes the earlier work on the energy-based discontinuous Galerkin method for second-order wave equations to fourth-order semilinear wave equations. We first rewrite the problem into a system with a second-order spatial derivative, then apply the energy-based discontinuous Galerkin method to the system. The proposed scheme, on the one hand, is more computationally efficient compared with the local discontinuous Galerkin method because of fewer auxiliary variables. On the other hand, it is unconditionally stable without adding any penalty terms, and admits optimal convergence in the $L^2$ norm for both solution and auxiliary variables. In addition, the energy-dissipating or energy-conserving property of the scheme follows from simple, mesh-independent choices of the interelement fluxes. We also present a stability and convergence analysis along with numerical experiments to demonstrate optimal convergence for certain choices of the interelement fluxes.
翻译:本文概括了先前关于以能源为基础的不连续 Galerkin 方法对二阶波方程式和四阶半线性波方程式所作的工作。 我们首先将问题改写成一个有二阶空间衍生物的系统, 然后对系统应用以能源为基础的不连续 Galerkin 方法。 与本地的不连续加列尔金方法相比,拟议办法一方面由于辅助变量较少,在计算上效率更高。 另一方面,它无条件稳定,不增加任何惩罚条件,并承认溶液和辅助变量在2美元标准上的最佳趋同。 此外,该计划的耗能性或节能性财产由简单的、中位的互通通通通量选择产生。 我们还提出一个稳定性和趋同性分析,同时进行数字实验,以显示对内流通量的某些选择的最佳趋同性。