双曲守恒律的整体Riemann解及其稳定性分析

项目名称： 双曲守恒律的整体Riemann解及其稳定性分析

项目编号： No.11271176

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 孙梅娜

作者单位： 鲁东大学

项目金额： 50万元

中文摘要： 非线性双曲守恒律Riemann问题的研究有着重要的理论意义和应用价值，本项目主要利用特征分析法和自相似的粘性消失法对几类双曲守恒律系统的Riemann问题进行研究。首先研究具有两个点火限的Majda-CJ模型的广义Riemann问题，并通过自相似的粘性消失法来检验其Riemann解的合理性。其次研究带有间断系数的双曲守恒律的Riemann问题和基本波的相互作用问题，并对其Riemann解在局部扰动下的稳定性进行分析。最后我们研究二维气体动力学方程组各种简化模型在三片常数情形下和在四片常数情形下的自相似Riemann问题以及非自相似的Riemann问题。上述研究内容都是双曲守恒律理论中最基本和最重要的问题之一，本项目的研究成果有望有效地推动双曲守恒律理论的发展，更深入的理解双曲守恒律弱解的间断特性和奇性的发展机制。

中文关键词： 双曲守恒律；燃烧模型；间断流；黎曼问题；整体解

英文摘要： It is important to study the Riemann problem for the nonlinear hyperbolic systems of conservation law in the theory and application. In this project, we will study the Riemann problem for several hyperbolic systems of conservation law by employing the characteristic analysis method and the self-similar viscosity vanishing approach. At first, we study the generalized Riemann problem for the Majda-CJ model with two ignition thresholds and we also check that whether the Riemann solutions are reasonable or not through the self-similar viscosity vanishing approach. Then, we study the Riemann problem and interactions of elementary waves for the hyperbolic conservation laws with discontinuous coefficients and analyze the stability of the Riemann solutions under the local small perturbation. Finally, we not only study the self-similar Riemann problem in which the initial states are three pieces of constants or four pieces of constants, but also study the non-selfsimilar Riemann problem for the simplified models of two-dimensional gas dynamics equation. The content of this research is fundamental and important for the theory of hyperbolic conservation laws. The research achievement is hoped to promote efficiently the development of the theory of hyperbolic conservation law and is helpful to understand the discontinuous p

英文关键词： hyperbolic conservation law；combustion model；discontinuous flux；Riemann problem；global solution