Numerical Gradient Schemes For Partial Difference Equations Based On The High-order Compact Difference Scheme

This paper involves several aspects of numerical methods, and it mainly focuses on the application on partial differential equations of numerical gradient which based on the compact difference scheme. At the same time, the paper talks about the application of Richardson extrapolation method on the partial differential equations involving time items.The numerical gradient method discussed in this article is essentially the combination of the collocation polynomial and the Hermite interpolation, and through that we obtain the expression of the numerical solutions at the intermediate points of the adjacent grid points. In fact, the expression relates to the expressions of the partial derivative and equation solution. The highlight of this paper is making the number of valid numerical points obtained double, but the time we use almost does not change. The Richardson extrapolation scheme is mainly used in the partial differential equation relating to time items, in order to improve the accuracy of errors.The main research work and innovation of this paper are as follows: The first one is numerical gradient scheme, which is based on Hermite interpolation and collocation polynomial. Zhi-Zhong Sun deduced a high-order compact difference scheme, which greatly improves the accuracy of the numerical solution error of partial differential equations. The scheme talked in the paper is based on this compact difference form. We will take up the advantages of the scheme, and then make up for its some problems. This process is as follows. Firstly, we get the numerical solutions, errors and error accuracy of the partial differential equations at the grid points by compact difference scheme. Then we obtain the partial derivatives of the numerical solutions at the grid points. Lastly, we get the numerical solutions at the intermediate points through Hermite interpolation. We get more numerical solutions but the space interval remains the same. The other is Richardson extrapolation scheme. The main function of Richardson extrapolation is to accelerate the convergence of the series. This article will apply it to the equations in order to improve the accuracy of the error.