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

我们对研究拟线性双曲抛物型偏微分方程的非线性波的性质很感兴趣。物理系统包括可压缩的Navier-Stokes和Euler方程，MHD和非线性弹性方程。我们也对非线性波的有限差分格式的计算感兴趣。基于非线性叠加和非线性波格林函数的方法在研究单空间变量PDE的非线性波时是有效的。例如，我们表明，冲击或多或少压缩比经典的气体动态冲击依赖于耗散变量在敏感的方式以前未被怀疑。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。在耗散格式和二维强、弱冲击的非线性稳定性和不稳定性的研究中取得了初步成功，这表明该方法适用于更一般的研究。发生在气体流动、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，同时存在的化学反应、电磁和其他效应使波的性质更加丰富。有几种不同类型的非线性相互作用的波。本提案的目的是研究这些非线性现象使用数学技术最近提出的提议，能够检测局部以及波相互作用的全局效应。对于我们的项目来说，重要的是理解这些问题的计算的数值格式。我们的方法开始为方案产生复杂的波动现象，这将帮助我们设计有效的方案来计算复杂的物理情况，如燃烧。* * *

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. ***