RUI: Adaptive High-Order Methods for Solving PDEs

本计画发展并分析求解偏微分方程式之网格与方法自适应高阶数值方法。也就是说，高阶方法，如谱方法，韦诺格式，径向基函数和离散变量展开将与各种网格尺寸相结合，以创建一个高精度，灵活和鲁棒的方法，根据该区域的光滑度的解决方案而变化。高阶自适应粘性间断Galerkin方法也将被认为是稳定的非线性问题，提高精度。这种技术可以被认为是最佳的，因为它们联合收割机最好的网格与最高可能的阶精度在整个计算域。在一个重要的应用中，一个有效的三维并行多域网格自适应谱罚代码将被创建用于模拟超燃冲压发动机凹腔火焰保持器中的反应流，这对设计超声速发动机至关重要。这项工作将使用一个已经成功的二维静态多域谱惩罚反应代码。一维和二维网格自适应混合方法将同时得到发展，这些鲁棒、精确和自适应的技术在燃烧、非均匀介质中的波传播和超声速冲击波等方面有着广泛的应用。这样的问题需要高精度和高效率的数值算法，以捕捉小规模的解决方案的功能。本项目中提出的技术非常适合执行这些类型的模拟。除了超音速超燃冲压发动机的例子外，一个有趣的应用是模拟燃料电池单元中的水传播。了解小尺度现象对于提高燃料电池的燃料效率至关重要。在这种情况下，自适应高阶数值模拟是必要的，因为实验室实验是昂贵的，难以测量，甚至无法捕捉这样的小尺度现象。最后，本项目推进了最先进的计算方法，并将为各种具有挑战性的物理和工程问题的调查提供有价值的见解。

This project develops and analyzes mesh and method-adaptive high-order numerical methodsfor solving partial differential equations. That is, high order methods such as spectralmethods, WENO schemes, radial basis functions and discrete variable expansions willbe incorporated with various mesh sizes to create a highly accurate, flexible and robustmethod that varies by domain depending on the smoothness of the solution in that region.High order adaptive viscosity discontinuous Galerkin methods will also be considered tostabilize the nonlinear problem and enhance accuracy. Such techniques can be consideredas optimal, since they combine the best mesh with the highest possible order accuracythroughout the computational domain. In one important application, an efficient threedimensional parallel multi-domain mesh-adaptive spectral penalty code will be created forthe simulation of the reactive flow in a scramjet engine's cavity flame holder, which iscritical for designing a supersonic engine. This work will use an already successful twodimensional static multi-domain spectral penalty reactive code. One and two dimensionalmesh-adaptive hybrid methods will be simultaneously developed.These robust, accurate and adaptive techniques have a broad range of applications,including combustion, wave propagation in heterogeneous media and supersonic shockwaves. Such problems demand highly accurate and efficient numerical algorithms in orderto capture small-scale features of the solution. The proposed techniques in this project arewell suited to perform these types of simulations. In addition to the supersonic scramjetengine example, one interesting application is the modeling of water propagation in afuel cell unit. Understanding the small-scale phenomenon is critical to enhance the fuelefficiency of a fuel cell. In this case, the adaptive high order numerical simulations arecrucial since the laboratory experiments are costly, difficult to measure and even fail tocapture such small-scale phenomenon. Finally, this project advances state of the artcomputational methods and will provide valuable insight into the investigation of variouschallenging physical and engineering problems.