Research And Applications Of High Order Compact Difference Method

Finite difference method is one of the most common and most widely used numerical ways of solving differential equations. With the development of modern science and engineering, the differential equations describing various problems are more complex, so high order compact difference scheme receives increasing atten-tion. Because of these advantages, such as high accuracy using considerably few grid points and spectral like-resolution of the boundary treatment, it has become increasingly popular in academia and industry.Currently, the fundamental idea of compact difference method is to construct one or several compact schemes for a given problem, which is easy to understand and thus easy to implement. However, this approach lacks flexible, such as we need to find different treatment for different problems and even for the same problem, differ-ent formats could be produced from different viewing angles:In order to overcome these disadvantages, a new method based on the finite volume has been developed: By choosing different type of dual partitions, we first integrate the differential equa-tion over the control volume; then combining with appropriate interpolation formula and integral formula, a family of high-order compact schemes are obtained. We will illustrate the availability of the proposed method through different models.1、A family of fourth-order and sixth-order compact difference schemes are present for two-dimensional and three-dimensional Poisson equation, respectively. Moreover, since the symmetry of the Laplace operator,we find that symmetrical and overlapping dual partitions are very useful to construct higher order schemes.2、Based on two type of dual partitions, a family of fourth-order compact difference schemes are derived for three-dimensional convection-diffusion equation with variable coefficients. However, due to the impact of the convective terms, it is difficult to build sixth-order schemes. To this end, we propose a new Richard-son extrapolation method, through which the sixth-order numerical solution can be obtained directly on the fine mesh.3、We discuss high-accuracy algorithms for both kinds three dimensional non-linear biharmonic equations:the original problems are split into two coupled sec-ond order elliptic differential equations by the introduction of intermediate variable v(=△u), this leads to a significant simplification of the problem. Since we set△u as a intermediate variables, it is much more difficult to solve the problem of first kind. Fortunately, combined compact difference scheme can greatly improve the accuracy of the approximate boundary.4、Finally, high-order compact ADI schemes are constructed for the three-dimensional time-fractional convection-diffusion equation:we first propose a trans-formation to eliminate the convection terms, then using the Pade approximation in space and the central differentiation in time, two high-order ADI schemes are derived based on two different splitting terms. Numerical experiments show that the second scheme is more accurate than the first one.