求解二维Euler方程的时一空守恒格式

NEW SPACE-TIME CONSERVATION SCHEMES FOR SOLVING 2-D EULER EQUATIONS

将作者原来得出的一维时－空守恒格式推广到了二维情形，得到了二维Ｅｕｌｅｒ方程的时．空守恒格式，并用几个典型算例进行了检验计算，结果表明：得到的二维时一空守恒格式保留了一维格式所有的优点，格式简单，通用性强，对微波等间断具有很高的分辨率．

The method of space-time conservation element and solution element (the CE/SEmethod, for short), developed by S.C. Chang[1], is a new numerical method which differs fromthe well-established methods and has many noniraditional features. Firstly, space and time areunified and treated on the same footing, and by the introduction of conservation element andsolution element, both local and global flux conservations in space and time instead of in spaceonly are enforced. Secondly, a zigzagging marching strategy in the space-time domain is em-ployed, such that flow information at each interface separating two conservation elements can beevaluated without interpolation or extrapolation. In particular, no Riemann solver is needed incalculating interfacial fluxes. Thirdly, The flow solution structure is not calculated through areconstruction procedure. Instead, the gradients of flow variables are solved simultaneously as independent unknowns. Finally, no approximation techniques other than Taylor's series expansion,no characteristic-based techniques and no directional splitting (in multiple spatial dimensions) areemployed in this method. So it is conceptually simple. It is capable of handling both continuousand discontinuous flows very well.But, the CE/SE method will become very complicated when it is extended to multi-dimensionalproblems. It is also difficult if we try to improve its accuracy. So, the CE/SE method is improved bythe author and a new constructing method is illtroduced. The resulting schemes not only have allthe features which the CE/SE method has, but also are much simpler and eajsier to use. Especiallyit is very easy to be extended to high dimensional cases. In this paper, the 1-D schemes developedin authors' previous works are extended to two dimensional problems. Here, two different definitions of space-time conservation elements (CEs) and solution elements (SEs) are introduced andthe corresponding space-time conservation schemes for solving 2-D Euler equations are derived.Numerical results of several typical flow problems show many advantages of the present schemessuch as its robustness, high efficiency, high accuracy and high shock resolution.