RUI: Adaptive High-Order Methods for Solving PDEs

本项目开发和分析了求解偏微分方程组的网格和方法自适应高阶数值方法。也就是说，高阶方法，如谱方法、WENO格式、径向基函数和离散变量展开，将结合不同的网格尺寸来创建一种高精度、灵活和健壮的方法，该方法随区域的光滑性而变化。高阶自适应粘性不连续Galerkin方法也将被考虑用于稳定非线性问题和提高精度。这种技术可以被认为是最优的，因为它们在整个计算域中结合了最好的网格和可能的最高阶精度。在一个重要的应用中，将创建一个高效的三维并行多区域网格自适应谱惩罚程序来模拟超燃冲压发动机腔体火焰稳定器内的反应流动，这对超音速发动机的设计是至关重要的。这项工作将使用一个已经成功的二维静态多区域谱惩罚反应代码。一维和二维网格自适应混合方法将被同时发展，这些稳健、精确和自适应的技术具有广泛的应用，包括燃烧、非均匀介质中的波传播和超音速激波。这类问题需要高精度和高效率的数值算法来捕捉解的小尺度特征。本项目中提出的技术非常适合执行这些类型的模拟。除了超音速超音速超燃冲压发动机的例子外，一个有趣的应用是模拟水在燃料电池单元中的传播。了解这种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至关重要的，因为实验室实验成本高，测量困难，甚至无法捕捉到这样的小尺度现象。最后，本项目发展了最先进的计算方法，并将为各种物理和工程问题的研究提供有价值的见解。

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.