论文标题
在等距和散射点上高阶不连续的Galerkin方法的稳定离散
Stable discretisations of high-order discontinuous Galerkin methods on equidistant and scattered points
论文作者
论文摘要
在这项工作中，我们提出并研究了等距和分散搭配点上不连续的盖尔金方法的稳定的高阶搭配型离散。我们通过将离散最小二乘正方形的概念纳入不连续的Galerkin框架中来做到这一点。离散的最小二乘近似值使我们能够在任意配置点上构建稳定和高阶精确近似值，而离散的最小二乘正方形规则使我们允许我们稳定且精确的数值集成。两种方法均通过使用离散正交多项式的碱基有效地计算。因此，所提出的离散化概括不连续的盖尔金方法的已知类别，例如不连续的Galerkin搭配频谱元素方法。我们能够证明拟议的离散化的保护和线性$ l^2 $稳定性。最后，数值测试研究了它们的准确性，并证明了它们扩展到两个空间维度中的非线性保护定律，系统，长期模拟以及可变系数问题。
In this work, we propose and investigate stable high-order collocation-type discretisations of the discontinuous Galerkin method on equidistant and scattered collocation points. We do so by incorporating the concept of discrete least squares into the discontinuous Galerkin framework. Discrete least squares approximations allow us to construct stable and high-order accurate approximations on arbitrary collocation points, while discrete least squares quadrature rules allow us their stable and exact numerical integration. Both methods are computed efficiently by using bases of discrete orthogonal polynomials. Thus, the proposed discretisation generalises known classes of discretisations of the discontinuous Galerkin method, such as the discontinuous Galerkin collocation spectral element method. We are able to prove conservation and linear $L^2$-stability of the proposed discretisations. Finally, numerical tests investigate their accuracy and demonstrate their extension to nonlinear conservation laws, systems, longtime simulations, and a variable coefficient problem in two space dimensions.
