A High-Efficient Algorithm For Non-Linear Dynamical System By The Continuous Finite Element Method

In this paper,we give our considerations in solving the non-linear dynamical systems by quadratic finite element method.Firstly,we use 4-stage explicit Runge-Kutta method to get the right nod value of the element,and then we propose a new formula to get the mid-nod value of the element by just giving a new linear combination of the K's which were calculated before.So we needn't calculate new function value.Lastly,we also prove that the new formula has three order of accuracy and this is the best we can get through 4-stage explicit Runge-Kutta method.In the light of this method we can also get similar formula for any nod in the element and they have the some precision as the mid nod formula we give.This means we can use finite element method of arbitrary polynomial order.For solving the non-linear equations which are discrete formulation of the non-linear differential equation by finite element method or interpolated coefficient finite element method,usually we use Newton method,but now with the mid nod and the right nod values in hand,together with the left nod value of the element we already know,we use this values as initial values for iterative algorithm,then we only need to use simple iterative method for the discrete nonlinear equations several times. This method saves a lot of computation time and quantity of calculation.We can also use extrapolation method to provide initial values.We calculate the first two elements with any kind method,from the third element on,we use the two elements(in all five points) just calculated immediately before this element.We construct a quartic interpolation polynomial and let it provides the mid nod and the right nod values of the next element.Numerical experiments indicate that the above method we proposed is effective.And this method can also be used for solving semi-linear parabolic equation.