Mathematical Sciences: High Order Methods for Time Dependent Equations

“含时方程的高阶方法”该项目需要研究真正含时偏微分方程组的长期积分。它涉及到高阶有限差分、有限元格式以及谱方法的发展和应用。所开发和应用的方法适用于具有复杂几何形状的多维空间问题。在开发和评估用于长期集成的高阶方法时，主要考虑的是它们的可伸缩性。在已有的ENO代码并行化经验的基础上，提出了基于谱和有限元的ENO代码并行化的进一步努力。研究分为理论和应用两个部分，这两个部分环环相扣。该理论是由正在调查的应用程序类型驱动的，而应用程序使用的是理论发展。研究表明，为了对复杂的随时间变化的流动进行数值模拟，高阶方法是必要的。然而，这些方法由于其高精度，健壮性较差。所提出的研究涉及高阶方法的成功实现问题。这既涉及理论问题(如解决100年前的吉布斯现象)，也涉及更多的应用问题。在应用方面，将研究激波和氢喷流相互作用的混合增强问题。这就需要模拟空气激波通过高马赫数的氢喷流。高阶方法在这里是强制性的，因为感兴趣的过程是喷嘴内部的混合和燃烧。目前的代码Spectral和ENO将进行修改，以处理更一般的情况。目前正在应用ENO和光谱法的并行版本，并将对其进行进一步修改。

"High Order Methods for Time Dependent Equations" The project entails research in long term integrations of genuinely time dependent partial differential equations. It involves the development and application of high order finite differences and finite element schemes as well as Spectral methods. The methods to be developed and applied are suitable for problems in several space dimensions with complicated geometries. A major consideration in developing and assessing high order methods for long term integrations is their scalability. Based on the experience gained already in parallelizing ENO codes, a further effort to parallelize spectral and finite element based ENO codes is proposed. The research has two components, theory and applications, those components are linked together. The theory is motivated by the type of applications under investigation and the applications use the theoretical developments. It has been shown that in order to numerically simulate complicated flows that change in time, high order methods are necessary. However those methods, because of their high accuracy, are less robust. The proposed research deals with the issue of the successful implementations of high order methods. This involved theoretical issues (as solving the 100 years old Gibbs Phenomenon) as well as more applied issues. On the applied side, the problem of mixing enhancement by interactions of Shock waves and hydrogen jets will be studied. This entails the simulation of an air shock passing through an hydrogen jet in high Mach numbers. High order methods are mandatory here since the process of interest is the mixing and combustion inside the jets. The current codes, Spectral and ENO are to be modified to handle more general cases. Parallel versions of ENO and spectral methods are being applied now and will be further modified.