The Semi-discrete Finite Element Method For The Coupled Equations

The collision process of cars is a short process. In the short process,the cars’interface will appear similar to the properties of Non-Newtonian fluid. Cauchy equation and P-T/T equation are employed to describe the change of cars’stress and velocity in the collision process.On the basis of theoretical knowledge of Sobolev space and the finite element method. The semi-discrete finite element method is used to study convergence of the coupled e-quations. Firstly, the component forms of the coupled equations are given. The space domain is discretized by 4-points bicubic Hermite shape function, so semi-discrete forms of Cauchy and P-T/T equations are obtained.Secondly, variational forms of coupled equations are obtained on the time. Then the time domain is discretized by finite difference scheme:Euler scheme and Crank-Nicolson scheme. The full discrete schemes of coupled equations are received, which is conducive to analyze convergence of coupled equations.Finally, it takes advantage of semi-discrete finite element method to study conver-gence of Cauchy equation. The space domain is discretized by 4-points bicubic Her-mite shape function. The Convergent order of the Cauchy equation is O(h2+(Δt)) and O(h2+(Δt)2) respectively, when Euler scheme and Crank-Nicolson scheme are employed on the time respectively. In numerical experiments, the error of the Cauchy equation is shown by comparing the numerical solutions with the exact solutions. Numerical results show that the convergence of Crank-Nicolson scheme is better than Euler scheme,which coexists with theoretical analysis.