论文标题
双曲线保护法的不连续盖尔金离散法中的整体凸限制
Monolithic convex limiting in discontinuous Galerkin discretizations of hyperbolic conservation laws
论文作者
论文摘要
在这项工作中，我们提出了一个框架，用于在不连续的Galerkin（DG）离散化中执行离散最大原则。开发的方案适用于标量保护法和双曲线系统。我们用于限制体积项的方法类似于最近提出的连续Galerkin近似方法，而DG通量术语则需要新颖的稳定技术。分段伯恩斯坦多项式用作DG空间的形状函数，从而促进了使用非常高阶的空间近似值。我们讨论了一种新的，可证明的不变域保存DG方案的设计，然后通过最新的子电池限制器扩展，以获得高阶保留近似值。限制过程可以在半混凝土设置中制定。因此，算法不会抑制与稳态溶液的收敛。我们提出了各种基准问题的数值结果。这项研究中考虑的保护定律是线性和非线性标量问题，以及气体动力学和浅水系统的Euler方程。
In this work we present a framework for enforcing discrete maximum principles in discontinuous Galerkin (DG) discretizations. The developed schemes are applicable to scalar conservation laws as well as hyperbolic systems. Our methodology for limiting volume terms is similar to recently proposed methods for continuous Galerkin approximations, while DG flux terms require novel stabilization techniques. Piecewise Bernstein polynomials are employed as shape functions for the DG spaces, thus facilitating the use of very high order spatial approximations. We discuss the design of a new, provably invariant domain preserving DG scheme that is then extended by state-of-the-art subcell flux limiters to obtain a high-order bound preserving approximation. The limiting procedures can be formulated in the semi-discrete setting. Thus convergence to steady state solutions is not inhibited by the algorithm. We present numerical results for a variety of benchmark problems. Conservation laws considered in this study are linear and nonlinear scalar problems, as well as the Euler equations of gas dynamics and the shallow water system.
