The convergence of the first-order hyperbolic partial differential equations in non-Newton fluid is analyzed. This paper uses coupled partial differential equations (Cauchy fluid equations, P-T/T stress equation) on a macroscopic scale to simulate the free surface elements. It generates watershed by excessive tensile elements. The semi-discrete finite element method is used to solve these equations. These coupled nonlinear equations are approximated by linear equations. Its super convergence is proposed.

In this paper, a dual-scaled method is introduced to simulate the nonlinear property of porous honeycomb material. In microscopic scale, stochastic analysis upon a detailed representation of the hexagonal cells is applied. In macroscopic scale, coupled fluid-solid PDEs with a modified stochastic item are used to describe the rheology of non-Newtonian property of honeycomb. Semi-discrete finite element method (FEM) is applied to solve the PDEs. Comparison of stochastic dynamic system with definite dynamic system is introduced. Numerical Results of Euler explicit time scheme and Crank-Nicolson semi-implicit time scheme are presented.