论文标题
化学反应，可压缩的Euler方程，具有阳性和熵结合的不连续的Galerkin方法。第一部分：一维情况
Positivity-preserving and entropy-bounded discontinuous Galerkin method for the chemically reacting, compressible Euler equations. Part I: The one-dimensional case
论文作者
论文摘要
在本文中，我们开发了一种完全保守的，具有积极性的且熵结合的不连续的盖尔金方案，用于模拟具有复杂热力学的化学反应，可压缩的Euler方程。所提出的公式是约翰逊和凯彻先前开发的保守，高阶数值方法的扩展[J.计算。 Phys。，423（2020），109826]，在相邻元素之间保持压力平衡。在我们两部分论文的第一部分中，我们专注于一维情况。我们的方法植根于通过对多组分可压缩欧拉方程的熵溶液所满足的最小熵原理，这是Gouasmi等人证明的。 [Esaim：数学。模型。 numer。肛门，54（2020），373--389]，用于非反应流。我们首先表明，最小熵原理也存在于反应情况下。接下来，我们介绍了溶液具有非负物种浓度，正密度，正压和有界熵所需的成分。我们还讨论了如何保留上述元素之间压力平衡的能力。操作员分裂用于处理僵硬的化学反应。为了确保对反应步骤中最小熵原理的满意，我们基于逐件求解熵的不连续的Galerkin方法开发了一种不连续的Galerkin方法，用于求解普通的微分方程。开发的配方用于计算规范的一维测试用例，即热气泡的对流，多组分冲击管流以及具有详细化学性质的移动氢氧气爆炸波。我们发现，在仅执行阳性特性时，与单个成分更大的范围相比，执行熵结合可以大大减少出现的大规模非线性不稳定性。
In this paper, we develop a fully conservative, positivity-preserving, and entropy-bounded discontinuous Galerkin scheme for simulating the chemically reacting, compressible Euler equations with complex thermodynamics. The proposed formulation is an extension of the conservative, high-order numerical method previously developed by Johnson and Kercher [J. Comput. Phys., 423 (2020), 109826] that maintains pressure equilibrium between adjacent elements. In this first part of our two-part paper, we focus on the one-dimensional case. Our methodology is rooted in the minimum entropy principle satisfied by entropy solutions to the multicomponent, compressible Euler equations, which was proved by Gouasmi et al. [ESAIM: Math. Model. Numer. Anal., 54 (2020), 373--389] for nonreacting flows. We first show that the minimum entropy principle holds in the reacting case as well. Next, we introduce the ingredients required for the solution to have nonnegative species concentrations, positive density, positive pressure, and bounded entropy. We also discuss how to retain the aforementioned ability to preserve pressure equilibrium between elements. Operator splitting is employed to handle stiff chemical reactions. To guarantee satisfaction of the minimum entropy principle in the reaction step, we develop an entropy-stable discontinuous Galerkin method based on diagonal-norm summation-by-parts operators for solving ordinary differential equations. The developed formulation is used to compute canonical one-dimensional test cases, namely thermal-bubble advection, multicomponent shock-tube flow, and a moving hydrogen-oxygen detonation wave with detailed chemistry. We find that the enforcement of an entropy bound can considerably reduce the large-scale nonlinear instabilities that emerge when only the positivity property is enforced, to an even greater extent than in the monocomponent, calorically perfect case.
