Mathematical Sciences: Study of Nonlinear Waves in Compressible Flows and Mechanics

9623025刘我们感兴趣的是研究拟线性双曲-抛物型偏微分方程解的非线性波的性态。物理系统包括可压缩的Navier-Stokes方程和Euler方程、MHD和非线性弹性方程。我们还对用于计算这些系统的有限差分格式的非线性波感兴趣。基于非线性叠加和非线性波的格林函数的方法被证明是研究一维空间变量偏微分方程组的非线性波的有效方法。例如，我们证明了比经典气体动力学激波更大或更小的压缩激波以一种以前未曾预料到的敏感方式依赖于耗散变量。我们打算推广和改进数值波的方法，以及在多个空间变量中的偏微分方程组。对二维弱激波和强激波的耗散格式和非线性稳定性与不稳定性的研究取得了初步成功，表明该方法适用于更一般的研究。发生在气流、太阳风、燃烧、固体物质中的物理现象往往是高度非线性的。材料的压缩、膨胀、剪切、弯曲都会产生波，而波的行为因化学反应、电磁和其他效应而变得更加丰富。有几种不同类型的波相互作用是非线性的。本方案的目的是利用作者最近提出的能够检测波相互作用的局部和全局效应的数学方法来研究这些非线性现象。对我们的项目来说，重要的是理解用于计算这些问题的数值格式。我们的方法开始为格式产生复杂的波动现象，这将帮助我们设计有效的格式来计算复杂的物理情况，如燃烧情况。***

9623025 Liu We are interested in studying the behaviour of nonlinear waves for quasilinear hyperbolic-parabolic partial differential equations. Physical systems include the compressible Navier-Stokes and Euler euqations, MHD, and nonlinear elasticity equations. We are also interested in the nonlinear waves for the finite difference schemes for the computations for these systems. The approach based on nonlinear superpositions and the Green functions for nonlinear waves has proved effective in studying the nonlinear waves for PDE in one space variable. For instance we show that shocks more or less compressive than the classical gas dynamic shocks depend on the dissipative variable in sensitive ways previously unsuspected. We intend to generalize and refine the approach to numerical waves and also to PDE in more than one space variable. Preliminary success in the study of dissipative schemes and nonlinear stability and instability of 2-dimensional weak and strong shocks indicates that the approach is suitable for more general studies. %%% Physical phenomena occur in gas flow, solar wind, combustions, solid materials are often highly nonlinear. Compression, expansion, shearing, bending of materials give rise to waves, whose behaviour is made richer by the chemical reactions, electro-magnetic and other effects that are also present. There are several different types of waves which interact nonlinearly. The goal of the present proposal is to study these noninear phenomena using mathematical technique recently introduced by the proposer, which are able to detect the local as well as global effects of wave interactions. Of importance to our project is the understanding of numerical schemes for the computations of these problems. Our approach is beginning to yield the intricate wave phenomena for the schemes which will help us to design effective schemes for computing complex physical situations such as that of combustions. ***