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摘要
The approximate inertial manifolds (AIMS) of Burgers equation is approached by nonlinear Galerkin methods, and it can be used to capture and study the shock wave numerically in a reduced system with low dimension. Following inertial manifolds, the asymptotic behavior of Burgers equation, an infinite dimensional dissipative dynamic systems, will evolve to a compact set known as a global attractor, which is finite-dimensional, and the nonlinear phenomena are included and captured in such global attractor. In the application, nonlinear Galerkin methods is introduced to approach such inertial manifolds. By this method, the solution of the original system is projected onto the complete space spanned by the eigen-functions or the modes of the linear operator of Burgers equation, and nonlinear Galerkin method splits the infinite-dimensional phase space into two complementary subspaces: a finite-dimensional one and its infinite-dimensional complement. Then, the post-processed Galerkin's procedure is used to approximate the solution of the reduced system, with the introduction of the interaction between lower and higher modes. Additionally, some numerical examples are presented to make a comparison between the traditional Galerkin method and nonlinear Galerkin method, in particular, some sharp jumping phenomena, which are related to the shock wave, have been captured by the numerical method presented. As the conclusion, it can be drawn that it is possible to completely describe the dynamics on the attractor of a nonlinear partial differential equation (PDE) with a finite-dimensional dynamical system, and the study can provide a numerical method for the analysis of the nonlinear continuous dynamic systems and complicated nonlinear phenomena in finite-dimensional dynamic system, whose nonlinear dynamics has been developed completely compared with infinite-dimensional dynamic system.
