论文标题
熵保护特性和高阶连续盖金近似标量保护法的熵稳定
Entropy conservation property and entropy stabilization of high-order continuous Galerkin approximations to scalar conservation laws
论文作者
论文摘要
本文介绍了针对标量保护法的高阶连续盖尔金（CG）有限元离散法的线性和非线性稳定程序的设计。我们证明标准CG方法是正方形熵的熵保守。通常，熵产生/耗散的速率取决于管理方程的残差以及有限元近似与熵变量的准确性。线性高阶稳定的包含在熵预算方程中产生附加的源/水槽。为了平衡每个细胞中的熵产生量，我们使用强制性双线性形式和无参数熵粘度系数构建熵分离元件贡献。熵稳定项是高阶的一致性，并且在实践中实现了最佳收敛行为。为了保护除熵稳定性之外的局部边界，我们使用了有限元解的伯恩斯坦基础表示和新的子电池通量限制程序。潜在的不平等约束确保了局部熵条件和局部最大原则的有效性。线性和非线性测试问题的数值结果说明了所提出的修改的好处。
This paper addresses the design of linear and nonlinear stabilization procedures for high-order continuous Galerkin (CG) finite element discretizations of scalar conservation laws. We prove that the standard CG method is entropy conservative for the square entropy. In general, the rate of entropy production/dissipation depends on the residual of the governing equation and on the accuracy of the finite element approximation to the entropy variable. The inclusion of linear high-order stabilization generates an additional source/sink in the entropy budget equation. To balance the amount of entropy production in each cell, we construct entropy-dissipative element contributions using a coercive bilinear form and a parameter-free entropy viscosity coefficient. The entropy stabilization term is high-order consistent, and optimal convergence behavior is achieved in practice. To enforce preservation of local bounds in addition to entropy stability, we use the Bernstein basis representation of the finite element solution and a new subcell flux limiting procedure. The underlying inequality constraints ensure the validity of localized entropy conditions and local maximum principles. The benefits of the proposed modifications are illustrated by numerical results for linear and nonlinear test problems.
