Blended High-order Compact Difference Schemes For Elliptic Equation

In this paper,two kinds of blended high-order compact difference schemes are constructed for the general elliptic equations.Firstly,based on the Taylor series expansion,a blended fourth-order compact difference scheme for solving the one-dimensional(1D)elliptic equation is deduced,Then,based on the fourth-order difference scheme,the blended sixth-order compact difference scheme is proposed to solve the 1D elliptic equation by Taylor's series expansion method.The truncation error of the two schemes are analyzed,respectively.The results show that the theoretical accuracy orders of the two kinds of difference schemes are fourth and sixth order,respectively.Finally,some numerical experiments are carried out by using some examples with exact solutions and the numerical results are compared with other schemes to show the superiority of the present schemes.For the two-dimensional(2D)problem,a fourth-order and a sixth-order blended compact difference schemes for solving the 2D elliptic equation are derived on the basis of 1D scheme.The truncation errors of the two difference schemes are analyzed,respectively.And then the numerical experiments are conducted to verify the accuracy of the present methods.For the three-dimensional(3D)problem,a blended fourth-order compact difference scheme is deduced and the truncation error of the scheme is analyzed,Finally,the numerical results are given and are compared with those by other schemes.The model equation studied in this paper is general,especially for the steady convection diffusion equation,the schemes developed in this paper can well simulate numerically problems with large Reynolds numbers,which is a major advantage of the present schemes compared to other schemes.