Mathematical Sciences: High Order Methods for Time Dependent Equations

点击翻译按钮获取中文摘要

"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.