High Order Methods for Linear and Nonlinear Waves

研究者和他的同事研究了线性和非线性波的高阶精度计算方法，重点是冲击波的计算和电磁波的模拟。工作包括高阶有限差分、有限元和谱方法的发展和分析，以及这些方法在计算流体动力学和计算电磁学中的应用。特别是，这些努力包括以下组成部分：激波计算的高阶方法，包括有限差分WENO格式，超音速反应流动的频谱方法和有限元不连续Galerkin方法；麦克斯韦方程组的高阶方法；和完美匹配的吸收层。计算机现在在工程和其他应用科学中得到更广泛的应用。有效地使用计算机的一个重要组成部分是设计和分析有效的数值算法。计算机辅助飞机设计等重要应用在很大程度上依赖于应用数学家设计的高效可靠的算法。研究人员在本项目中开发和分析的高阶方法将显著提高关键应用领域（如航空航天工业、通信和材料科学）中使用的算法的效率和可靠性。

Shu0207451 The investigator and his colleague study high-order-accuracycomputational methods for linear and nonlinear waves, withemphasis on shock wave calculations and simulation ofelectro-magnetics waves. The work includes the development andanalysis of high-order finite difference, finite element, andspectral methods, as well as applications of these methods tocomputational fluid dynamics and computational electro-magnetics.In particular, the efforts include the following components:high-order methods for shock wave calculations, including finitedifference WENO schemes, spectral methods for supersonic reactiveflows, and finite element discontinuous Galerkin methods;high-order methods for Maxwell's equations; and perfectly matchedabsorbing layers. Computers are now used more extensively in engineering andother applied sciences. An important component to effectivelyuse computers is the design and analysis of efficient numericalalgorithms. Important applications such as computer-aided designof aircraft are to a large extent dependent on efficient andreliable algorithms designed by applied mathematicians. Thehigh-order methods the investigators develop and analyze in thisproject should significantly increase the efficiency andreliability of algorithms used in crucial application areas suchas aerospace industry, communications, and material science.