The Construction And Application Of Several High Accuracy Compact Difference Schemes

The characteristics of compact difference schemes and super compact difference schemes are the high accuracy and high performance in numerical calculation, but there are some imperfections while the schemes are used for solving the actual problems.Firstly, in many actual fluid flow problems, the effects come from upstream will be more than that come from downstream. So it is inappropriate that the effects come from upstream and downstream are considered as the same when the symmetrical super compact difference schemes are used to solve numerically the actual fluid flow problems.Secondly, the great amount of uniform grid points are employed to achieve the higher accurate numerical results, it will greatly increase the computational efforts. So the non-uniform grid points are important for solving the actual fluid problem.In this thesis, an upwind super compact difference scheme(USCD) is presented based on the compact difference scheme and Taylor expansion, a third-point fourth-order compact difference scheme(N-CD4) and a third-point sixth-order super compact difference scheme(N-CSCD) for the non-uniform grid are also presented.The numerical characteristics of the upwind super compact difference scheme(USCD) is analyzed by using Fourier analysis, and compared with other upwind difference schemes and upwind compact difference schemes. According to analysis,USCD has better resolution and lower dissipation. The numerical solutions of the Burgers,KdV-Burgers and two-dimensional Burgers’ equations show that the USCD scheme has high-order accuracy and is effective for long time evolution.The third-point fourth-order compact difference scheme(N-CD4) and the third-point sixth-order super compact difference scheme(N-CSCD) are simple in form,flexible in grid subdivision and general in use. The truncation errors of the schemes are analyzed. The Burgers equation and the convection equation are solved numerically by using present scheme on the non-uniform grid, the numerical solutions are compared with those obtained from the standard third-point fourth-order compact difference scheme and third-point sixth-order super compact difference scheme on the uniform grid, and the results show that the present scheme have higher accuracy for large gradient problems.