Optimized Fourth-order Tridiagonal Compact Schemes And Their Initial Applications In Different Grid Systems

With the development of the science and computer technology, numerical simulationhas been widely used in fluid mechanics, electromagnetism, pneumatic acoustics, aerody-namics, atmospheric dynamics and so on. The wide applications prompt us to developnumerical schemes with higher order accuracy with high resolutions. Among the numer-ical schemes, compact finite diference scheme is a good choice due to its high order ofaccuracy and high resolution. Moreover, compared with the traditional diference schemeswhich are expressed by the linear combination of function values on several grids. Compactdiference schemes are expressed equivalent with a linear combination of derivatives in theleft hand side and a linear combination of function values in the right hand side. Thus,the compact diference scheme has higher order of accuracy under the condition of reducedgrid stamps. An important work in numerical mathematics is to obtain the more accuratenumerical solution. This is also the main work of this thesis, which can be ascribed as thefollowing three aspects:Firstly, based on the idea of modified wavenumbers should be close to the exactwavenumbers as large as possible, an optimal tridiagonal fourth-order compact diferencescheme and interpolation scheme on the staggered grid system are proposed in this thesis.Although it’s accuracy order is4, it has higher resolutions and preserves the characteristicsof group velocity. Numerical simulations showed that the maximum resolvable wavenum-bers of the optimal compact diference (interpolation) scheme is0.86π (0.63π). The groupvelocity can be preserved for wavenumber less than0.75π. All these values are larger than those obtained from standard compact schemes with fourth or sixth order. The optimalscheme, standard fourth and sixth order compact scheme schemes are employed to calculatethe first derivation and the propagations of small scale waves. The results show that theoptimal scheme is superior to other two schemes in the respects of reducing computationalerror and preserving group velocity.Secondly, based on the ideas of the polynomial fitting numerical boundary scheme(SFEBS)and the Taylor expansion boundary scheme (TEBS), a fourth-order numerical boundaryscheme (SF-TEBS4) for the optimized fourth-order staggered tridiagonal compact difer-ence scheme (OCS4) and its interpolation scheme (OCI4) on the staggered grid systemis proposed. The matrix eigenvalue of the combining scheme shows that, OCS4schemecombined with OCI4scheme and the fourth-order numerical boundary scheme SF-TEBS4can meet the requirements of the asymptotic stability. Numerical experiments shows thatthe global accuracy of the combination of all these schemes is fourth order. Moreover,SF-TEBS4can preserve group velocity. It can efectively reduce the growth rate of thecomputational error. This illustrates the combination of these schemes has higher numer-ical asymptotic stability.Thirdly, an optimized fourth-order compact finite diference scheme (OC4) has beendeveloped by us before. To illustrate the advantages of the OC4scheme on the aspects ofhigh spectral resolution and the property of keeping group velocity, we extend the appli-cation of the OC4scheme into two-dimensional problems and compare it with standardfourth order (SC4), sixth order (SC6) and eighth order (SC8) compact finite diference schemes. Two numerical examples are performed to illustrate this point. One is to simu-late the propagation of a wave packet and shows that OC4reduce84%of the error and cankeep group velocity for a long time when compared with SC4, SC6and SC8. The otheris to simulate the propagation of a Gaussian wave and shows that OC4reduce error morethan18%. These comparisons show that the OC4scheme solves two-dimensional problemsmore exactly and are more suitable for solving the convection of the two-dimensional smallscale waves.