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Shock capturing by anisotropic diffusion oscillation reduction

机译：通过各向异性扩散振荡的减少来捕捉冲击

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摘要

This paper introduces an anisotropic diffusion oscillation reduction (ADOR) scheme for shock wave computations. The connection is made between digital image processing, in particular, image edge detection, and numerical shock capturing. Indeed, numerical shock capturing can be formulated on the lines of iterative digital edge detection. Various anisotropic diffusion and super diffusion operators originated from image edge detection are proposed for the treatment of hyperbolic conservation laws and near-hyperbolic hydrodynamic equations of change. The similarity between anisotropic diffusion and artificial viscosity is discussed. Physical origins and mathematical properties of the artificial viscosity are analyzed from the point of view of kinetic theory. A form of pressure tensor is derived from the first principles of the quantum mechanics. Quantum kinetic theory is utilized to arrive at macroscopic transport equations from the microscopic theory. Macroscopic symmetry is used to simplify pressure tensor expressions. The latter provides a basis for the design of artificial viscosity. The ADOR approach is validated by using (inviscid) Burgers' equation, the gas tube problems, the incompressible Navier-Stokes equation and the Euler equation. Both standard centra difference schemes and a discrete singular convolution algorithm are utilized to illustrate the approach. Results are compared with those of third-order upwind scheme and essentially non-oscillatory (ENO) scheme.
机译：本文介绍了一种各向异性的扩散振荡降低（ADOR）方案，用于冲击波计算。在数字图像处理（尤其是图像边缘检测）和数字震荡捕获之间建立了联系。确实，可以在迭代数字边缘检测的线上制定数字冲击捕获。提出了各种源自图像边缘检测的各向异性扩散算子和超扩散算子，用于处理双曲守恒律和近双曲流体动力学方程。讨论了各向异性扩散与人工黏度之间的相似性。从动力学理论的角度分析了人工粘度的物理起源和数学性质。压力张量的一种形式是从量子力学的第一原理中得出的。利用量子动力学理论从微观理论得出宏观输运方程。宏观对称用于简化压力张量表达式。后者为人工粘度的设计提供了基础。 ADOR方法通过使用（无粘性）Burgers方程，煤气管问题，不可压缩的Navier-Stokes方程和Euler方程进行验证。标准中心差方案和离散奇异卷积算法均用于说明该方法。将结果与三阶迎风方案和基本非振荡（ENO）方案的结果进行比较。

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