Abstract Dynamic mesh adaptation methods require suitable refinement indicators. In the absence of a comprehensive error estimation theory, adaptive mesh refinement (AMR) for nonlinear hyperbolic conservation laws, e.g. compressible Euler equations, in practice utilizes mainly heuristic smoothness indicators like combinations of scaled gradient criteria. As an alternative, we describe in detail an easy to implement and computationally inexpensive criterion built on a two-level wavelet transform that applies projection and prediction operators from multiresolution analysis. The core idea is the use of the amplitude of the wavelet coefficients as smoothness indicator, as it can be related to the local regularity of the solution. Implemented within the fully parallelized and structured adaptive mesh refinement (SAMR) software system AMROC (Adaptive Mesh Refinement in Object-oriented C++), the proposed criterion is tested and comprehensively compared to results obtained by applying the scaled gradient approach. A rigorous quantification technique in terms of numerical adaptation error versus used finite volume cells is developed and applied to study typical two- and three-dimensional problems from compressible gas dynamics. It is found that the proposed multiresolution approach is considerably more efficient and also identifies – besides discontinuous shock and contact waves – in particular smooth rarefaction waves and their interaction as well as small-scale disturbances much more reliably. Aside from pathological cases consisting solely of planar shock waves, the majority of realistic cases show reductions in the number of used finite volume cells between 20 to 40%, while the numerical error remains basically unaltered.

中文翻译：

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多分辨率分析作为有效动态网格自适应的标准——SAMR 框架 AMROC 中欧拉方程的案例研究

摘要 动态网格自适应方法需要合适的细化指标。在缺乏综合误差估计理论的情况下，非线性双曲守恒定律（例如可压缩欧拉方程）的自适应网格细化 (AMR) 在实践中主要利用启发式平滑指标，如缩放梯度标准的组合。作为替代方案，我们详细描述了一种易于实现且计算成本低的标准，该标准建立在两级小波变换上，该变换应用了多分辨率分析中的投影和预测算子。核心思想是使用小波系数的幅度作为平滑度指标，因为它可以与解的局部规律有关。在完全并行化和结构化的自适应网格细化 (SAMR) 软件系统 AMROC（面向对象 C++ 中的自适应网格细化）中实施，所提出的标准经过测试并与应用缩放梯度方法获得的结果进行综合比较。就数值自适应误差与使用的有限体积单元而言，开发了一种严格的量化技术，并将其应用于研究可压缩气体动力学中典型的二维和三维问题。发现所提出的多分辨率方法效率更高，并且还可以更可靠地识别 - 除了不连续的冲击波和接触波 - 特别是平滑的稀疏波及其相互作用以及小规模干扰。除了仅由平面冲击波组成的病理情况外，