Mathematical Sciences: High Order Methods for Time Dependent Equations

“高阶方法的时间依赖方程”该项目需要研究长期的积分真正的时间依赖偏微分方程。它涉及高阶有限差分和有限元格式以及谱方法的发展和应用。待开发和应用的方法适用于具有复杂几何形状的多个空间维度的问题。在开发和评估长期积分的高阶方法时，一个主要考虑因素是它们的可扩展性。基于已经在并行化ENO代码中获得的经验，提出了进一步的努力，并行化基于谱和有限元的ENO代码。本研究由理论和应用两部分组成，这两部分是相互联系的。 该理论的动机是调查中的应用程序的类型和应用程序使用的理论发展。 为了数值模拟随时间变化的复杂流动，高阶方法是必要的。然而，这些方法由于其高精度而不太稳健。所提出的研究涉及高阶方法的成功实现的问题。 这涉及理论问题（如解决100岁的吉布斯现象）以及更多的应用问题。在应用方面，将研究通过激波和氢射流的相互作用来增强混合的问题。这就需要模拟空气冲击波在高马赫数下穿过氢射流。高阶方法在这里是强制性的，因为感兴趣的过程是射流内部的混合和燃烧。目前的代码，光谱和ENO将被修改，以处理更一般的情况。 ENO和谱方法的并行版本现在正在应用，并将进一步修改。

"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.