论文标题
浅水流的完全平衡的熵受控的不连续的盖金光谱元素方法：全局通量正交和细胞熵校正
Fully well-balanced entropy controlled discontinuous Galerkin spectral element method for shallow water flows: global flux quadrature and cell entropy correction
论文作者
论文摘要
我们提出了一种使用不连续的Galerkin光谱元素方法来解决浅水方程的新方法。我们建议的方法具有三个主要功能。首先，它具有一种类似于例如[20]。如参考所述，我们的方案不需要对稳定平衡的任何A-Priori知识，此外，它不涉及任何局部辅助问题的明确解决方案来近似此类均衡。该方案也是任意高阶，并验证了时间单元熵相等的连续。一旦将附加耗散添加到该方法中，后者就会成为不平等。该方法是从全局通量方法开始构建的，在该方法中，将额外的通量项构建为源的原始术语。我们表明，在节点光谱有限元元素的背景下，可以将其转化为对源术语积分的简单修改。我们证明，当使用高斯 - 洛巴托节点有限元元素时，这种修改的积分在稳态上等效于应用于磁通量的高阶高斯搭配方法。该方法在搭配点上是超浓缩的，因此在精神上提供了与[20]中提出的一种非常相似的良好平衡属性，尽管不需要明确计算稳态的局部近似值。为了控制熵产生，我们在细胞水平上引入人造粘度校正，并将其纳入该方案。我们提供了这些校正的准确性和平衡保存的理论和数值特征。通过广泛的数值基准测试，我们验证了我们的理论预测，稳态的准确性有了显着提高，并增强了鲁棒性的更复杂场景
We present a novel approach for solving the shallow water equations using a discontinuous Galerkin spectral element method. The method we propose has three main features. First, it enjoys a discrete well-balanced property, in a spirit similar to the one of e.g. [20]. As in the reference, our scheme does not require any a-priori knowledge of the steady equilibrium, moreover it does not involve the explicit solution of any local auxiliary problem to approximate such equilibrium. The scheme is also arbitrarily high order, and verifies a continuous in time cell entropy equality. The latter becomes an inequality as soon as additional dissipation is added to the method. The method is constructed starting from a global flux approach in which an additional flux term is constructed as the primitive of the source. We show that, in the context of nodal spectral finite elements, this can be translated into a simple modification of the integral of the source term. We prove that, when using Gauss-Lobatto nodal finite elements this modified integration is equivalent at steady state to a high order Gauss collocation method applied to an ODE for the flux. This method is superconvergent at the collocation points, thus providing a discrete well-balanced property very similar in spirit to the one proposed in [20], albeit not needing the explicit computation of a local approximation of the steady state. To control the entropy production, we introduce artificial viscosity corrections at the cell level and incorporate them into the scheme. We provide theoretical and numerical characterizations of the accuracy and equilibrium preservation of these corrections. Through extensive numerical benchmarking, we validate our theoretical predictions, with considerable improvements in accuracy for steady states, as well as enhanced robustness for more complex scenarios
