Second-Order Invariant Domain Preserving Approximation of the Euler Equations Using Convex Limiting
Second-Order Invariant Domain Preserving Approximation of the Euler Equations Using Convex Limiting
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DOI：
10.1137/17m1149961
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发表时间：
2017-10
期刊：
影响因子：
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通讯作者：
J. Guermond;Murtazo Nazarov;B. Popov;I. Tomas
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文献类型：
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作者：
J. Guermond;Murtazo Nazarov;B. Popov;I. Tomas
引入了一种新的二阶方法，用于近似可压缩的Euler方程。该方法保留了Euler系统的所有已知不变域：密度的阳性，内部能量的阳性和特定熵的局部最小原理。该技术结合了使用图粘度（GMS-GV1）的一阶，不变的域保证，保证最大速度方法，违反了不变的域，但熵一致，高阶方法。使用GMS-GV1方法自然产生的辅助状态的不变域用于定义高阶方法的局部边界，然后将其通过凸限制过程制成不变域。数值测试在最大规范中证实了新的GMS-GV2方法的二阶精度，其中2代表二阶。提出的凸限是通用的，可以应用于其他近似技术和其他双曲系统。
A new second-order method for approximating the compressible Euler equations is introduced. The method preserves all the known invariant domains of the Euler system: positivity of the density, positivity of the internal energy and the local minimum principle on the specific entropy. The technique combines a first-order, invariant domain preserving, Guaranteed Maximum Speed method using a Graph Viscosity (GMS-GV1) with an invariant domain violating, but entropy consistent, high-order method. Invariant domain preserving auxiliary states, naturally produced by the GMS-GV1 method, are used to define local bounds for the high-order method which is then made invariant domain preserving via a convex limiting process. Numerical tests confirm the second-order accuracy of the new GMS-GV2 method in the maximum norm, where 2 stands for second-order. The proposed convex limiting is generic and can be applied to other approximation techniques and other hyperbolic systems.