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

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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. ***