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

小行星9623025 本文主要研究拟线性双曲-抛物型偏微分方程的非线性波的行为。 物理系统包括可压缩Navier-Stokes方程和Euler方程、MHD方程和非线性弹性方程。我们也有兴趣在这些系统的计算有限差分格式的非线性波。基于非线性叠加和非线性波的绿色函数的方法被证明是研究单空间变量偏微分方程非线性波的有效方法。例如，我们表明，冲击或多或少压缩比经典的气体动力学冲击依赖于耗散变量的敏感方式以前未知的。我们打算推广和完善的方法，数值波和PDE在一个以上的空间变量。在耗散格式和二维弱激波、强激波的非线性稳定性和不稳定性研究中的初步成功表明，该方法适用于更一般的研究。 物理现象发生在气体流动、太阳风、燃烧、固体物质中，往往是高度非线性的。材料的压缩、膨胀、剪切、弯曲引起波，而化学反应、电磁和其他效应也使波的行为更加丰富。 有几种不同类型的波，它们非线性地相互作用。本提案的目标是使用提议者最近引入的数学技术来研究这些非线性现象，这些数学技术能够检测波相互作用的局部和全局效应。 我们的项目的重要性是理解这些问题的计算数值方案。我们的方法开始产生复杂的波动现象的计划，这将有助于我们设计有效的计划，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. ***