The Cubic Finite Volume Element Methods For Two Kinds Of Partial Differential Equations

In this paper,according to the characteristics of the equations,cubic finite volume element numerical schemes for an one-dimensional convection diffusion reaction equation and Burgers'equation are constructed respectively.The analyses of the convergence are given,and the validity of the theoretical analysis results and the numerical schemes are confirmed through numerical examples.The concrete research contents are as follows.In the first part,the optimal stress points are used to construct the dual partitions.Based on the space of piecewise cubic Lagrange interpolation trial function and the space of piecewise constant test function,a Crank-Nicolson cubic finite volume element scheme for a convection-diffusion reaction equation is constructed.The L~2norm error estimation of numerical solution is obtained.Furthermore,a scheme with fourth order accuracy both in the temporal and spatial directions is constructed by applying the Richardson's extrapolation method in the time direction.Numerical examples are given to verify the validity of the theoretical analysis results and the effectiveness of the proposed algorithm.In addition,the numerical example confirms that the derivative of numerical solution has fourth order super-convergence accuracy at the optimal stress points.In the second part,the Hopf-Cole transformation is used to transform Burgers'e-quation into a linear problem.And then the Crank-Nicolson cubic finite volume element scheme is constructed by using the similar approach given above.The L~2norm error estimation of the numerical solution for the linear problem is proved.At the same time,a Richardson's extrapolation method is used in the time direction to construct a scheme with fourth order accuracy both both in the temporal and spatial directions.Finally,nu-merical examples are provided to show feasibility of the method proposed in this paper.