Conservative and Stable Degree Preserving SBP Operators for Non-conforming Meshes
Conservative and Stable Degree Preserving SBP Operators for Non-conforming Meshes
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DOI：
10.1007/s10915-017-0563-z
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发表时间：
2016-11
影响因子：
2.5
通讯作者：
Lucas Friedrich;D. C. D. R. Fernández-D.-C.-D.-R.-Fernández-145704654;A. R. Winters;G. Gassner;D. Zingg;Jason E. Hicken
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文献类型：
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作者：
Lucas Friedrich;D. C. D. R. Fernández-D.-C.-D.-R.-Fernández-145704654;A. R. Winters;G. Gassner;D. Zingg;Jason E. Hicken
非协调数值近似为在计算域的局部区域或接近复杂的几何形状中需要高分辨率的应用提供了更大的灵活性。适用于实际应用的非协调方法的两个关键性质是守恒性和能量稳定性。由于能量稳定性和守恒性的证明可以离散地模拟偏微分方程组的连续分析，某些有限差分和间断Galerkin方法所具有的分部求和(SBP)性质对于双曲型守恒律的数值逼近是成功的。此外，SBP方法可以开发出高精度的方法，这对于包含多个空间和时间尺度的模拟是有用的。然而，现有的非协调SBP格式导致格式的整体程度降低，从而导致解误差的阶数降低。这种阶数的损失是由于通过同时近似项(SAT)的特定界面耦合造成的。在这项工作中，我们提出了一类新的SBP-SAT算子，它保持守恒性、能量稳定性，并且在非协调近似下不损失格式的次数。与现有的有限差分SBP算子的范数矩阵是精确的不同，新的保度离散化要求SBP算子的范数矩阵是一个次。通过严格的数学分析和数值验证，我们证明了新格式的基本性质。
Non-conforming numerical approximations offer increased flexibility for applications that require high resolution in a localized area of the computational domain or near complex geometries. Two key properties for non-conforming methods to be applicable to real world applications are conservation and energy stability. The summation-by-parts (SBP) property, which certain finite-difference and discontinuous Galerkin methods have, finds success for the numerical approximation of hyperbolic conservation laws, because the proofs of energy stability and conservation can discretely mimic the continuous analysis of partial differential equations. In addition, SBP methods can be developed with high-order accuracy, which is useful for simulations that contain multiple spatial and temporal scales. However, existing non-conforming SBP schemes result in a reduction of the overall degree of the scheme, which leads to a reduction in the order of the solution error. This loss of degree is due to the particular interface coupling through a simultaneous-approximation-term (SAT). We present in this work a novel class of SBP–SAT operators that maintain conservation, energy stability, and have no loss of the degree of the scheme for non-conforming approximations. The newdegree preservingdiscretizations require an ansatz that the norm matrix of the SBP operator is of a degree, in contrast to, for example, existing finite difference SBP operators, where the norm matrix isaccurate. We demonstrate the fundamental properties of the new scheme with rigorous mathematical analysis as well as numerical verification.