Optimized Compact Finite Difference Scheme And Its Initial Application

The research and application of compact fnite diference scheme with high order ofaccuracy have received much attentions in recent years. This kind of scheme plays impor-tant roles in scientifc computations. As the increasing complexity scientifc computations,the requirement on the numerical scheme is increasing also. Thus, the numerical schemeswith lower order do not fulfll the requirement. As a result, it is necessary to develop highorder and high accuracy numerical scheme.The content of this thesis can be ascribed to the following three points:First, based on the idea of diference schemes pseudo-wavenumbers should resolvethe physical wavenumbers as accurate high as possible we proposed an optimal tridiagonalcompact fnite diference scheme with fourth order accuracy and high resolution. On the onehand, the optimal compact scheme can be solved efciently by the algorithms which we havedeveloped recently to solve the (cyclic) tridiagonal equations. Thus, the frst derivation ofpartial diferential equations can be solved efciently by the optimized compact diferencescheme efcienty; On the other hand, the maximum of pseudo-wavenumber can be resolvedby the optimal scheme is2.5761, and larger than the pseudo-wavenumber of1.13097,which can be resolved by the traditional fourth order compact schemes. Therefore, theoptimized compact diference scheme is more appropriate to resolve small scale waves influid dynamics and aero-acoustic dynamics.Second, comparisons among the optimized fourth order compact fnite diference scheme, sixth and eighth order compact diference are performed. The comparative results showedthat the optimized fourth order scheme with low accuracy has higher resolution than thatof the sixth and eight order schemes. Therefore, we have to bear in mind that the accuracyand the resolution are two diferent concepts and should not be replaced with each other.In contrast, one should choose a scheme with appropriate order of accuracy and resolutionaccording to the problem needed to be solved.Third, compared with traditional fourth-order compact fnite diference scheme, theoptimized fourth order compact fnite diference scheme has high resolution and is moresuitable to resolve small scale fluctuation problems in practical applications. Numerical ex-periments illustrated that: although the accuracy of the optimized scheme is fourth-order,it has a smaller error than that of the traditional fourth-order compact fnite diference,especially for the small scale fluctuations. Our numerical experiments also showed theadvantages of the optimized fourth-order compact fnite diference scheme in simulatingthe wave propagation problem. Thus, according to the theoretical analysis and numericalexperiment results, we conclude that optimized compact fnite diference scheme is moreappropriate to resolve small scale fluctuations problems.