The Studies On Geometric Conservation Law For High Order Finite Difference Method

As one of the most antique numercial methods, Finite Difference Method (FDM) wins the favour of beginners by reason of its simpleness and understandability, conse-quently, it has become an essential matter of many textbooks on numerical methods. Nev-ertheless, few applications of FDM can be found in practical engineering problems, which confuses numerous researchers on Computational Fluid Dynamics (CFD), though FDM plays important roles of establishing the basic concepts and enlightening ideologies for nu-merical methods. In solid mechanics Finite Element Method (FEM) is working tolerably well, and almost all commercial softwares in fluid mechanics are constituted in the frame of Finite Volume Method (FVM), only few researchers write their computing codes based on FDM, and combined with high order finite difference schemes so it can be applied to study the mechanism of a flow around a simply configuration.At the present time, it is too difficult to simulate massive engineering problems with FDM, the primary reason is that FDM is not robust enough to handle complex mesh. Unlike FVM, for FDM the essential prerequisite is the strict establishment of coordi-nate transformation (from Cartesian coordinates to computational coordinates) and its in-verse transformation (from computational coordinates to Cartesian coordinates), so FDM strongly depends on coordinate transformation. Among all characteristics of coordinate transformation, the Geometric Conservation Law (GCL) can crucially affect computa-tional robustness of FDM, which is indicated by a mass of studies. The GCL of coordi-nate transformation comes into existence automatically for differential operation, but this is not true for difference operation. The difficulty to ensure GCL under difference oper-ation badly affects the computational robustness of FDM, and restricts its application in engineering problems.This paper detailedly analyses the sufficient condition to ensure GCL:Conserva-tive Metric Method (CMM), under the guidance of CMM the Symmetrical Conservative Metric Method (SCMM) is proposed so that each metric can represent its own geometric meanings when discretized by any difference operation, and subsequently the validity of SCMM is obviously validated by some numerical experiments on randomized grids. To analyse the influence of numerical results induced by GCL error, the impact of numerical convergence rate of GCL error caused by mesh smoothness is analysed firstly in this pa-per, and subsequently the influence of convergence rate of numerical results induced by GCL error is analysed, and all the analyses above is validated correspondingly by numer-ical experiments. The investigation in this paper shows that the maintenance of GCL can effectively reduce discretization error of numerical results, and if the mesh is not smooth enough for discretization schemes, maintaining GCL can increase the convergence rate of numerical results by a whole order.When the geometric meanings of different computational forms of metric are distin-guished, it is observed that different forms of Jacobian can represent different numerical volumes when discretized. If the computational cell is a parallelepiped, all the three nu-merical volumes represented by three different forms of Jacobian correspondingly are the same; on the other side if the cell is not a parallelepiped, the three numerical volumes will be different in a general way. Generally speaking, if the aberration of a computational cell to a parallelepiped is large, the difference of all three numerical volumes discretized by difference operation will not be small. Thus a mesh detect method based on SCMM is proposed to examine the quality of structured mesh. It is different from the mesh detect methods owned by commercial softwares, the detect method based on SCMM is corre-lated with the discretization scheme. On the same mesh, if the discretization schemes are different the indicators of the detect method based on SCMM will be different, thus the correlation of mesh quality with discretization scheme can be adequately materialized. At the end of this paper, the capability to simulate flowfield around three-dimensional complex configurations is achieved elementarily based on the studies of GCL.This dissertation is divided into seven chapters as follows:The first chapter is the introduction. The history and current status of FDM (espe-cially high order FDM) are reviewed briefly, and it is pointed out that hanging in doubt of GCL crucially restricts its application in engineering problems. Subsequently, a literature survey of GCL within finite difference frame is presented, and so as Volume Conservation Law (VCL) and Surface Conservation Law (SCL). At the end of this chapter, the studies aimed at GCL of this paper are introduced in brief.The GCL problem is proposed in the second chapter. Firstly the coordinate transfor-mation in FDM is introduced, with which the governing equation of generic conservation law can be transformed from Cartesian coordinates to computational coordinates. When the generic conventional characteristic of finite difference operators in computational co-ordinates has been presented, the GCL residual under finite difference operation can be deduced by freestream preservation property.The GCL is studied theoretically in the third chapter. When various solutions for GCL have been analysed simply, based on the sufficient condition to ensure GCL (CMM) the Symmetrical Conservative Metric Method (SCMM) is proposed to discretize metrics and Jacobian with finite difference operator. The validity of SCMM is obviously vali-dated by some numerical experiments on randomized grids, the numerical tests for both linear and nonlinear problems show that SCMM proposed in this paper can effectively reduce discretization error of numerical results with finite difference schemes, especially for nonlinear problems.The fourth chapter provides the study of the impact of numerical convergence rate of GCL error caused by mesh smoothness. The relationship between GCL error (when the inner and outer difference operators of metric are different) and mesh smoothness is theo-retically deducted, which is validated to be true by numerical experiments. In addition, if the convergence rates of numerical errors in different locations are different, how the sta-tistical error all over the whole flowfield behaves is analysed theoretically and detailedly.The relationship between GCL error and convergence rate of numerical error is stud-ied in the fifth chapter. Whe mesh is different degree of smoothness, how GCL error can impact the convergence rate of numerical error is analysed theoretically and numerically. The maintenance of GCL can effectively reduce discretization error of numerical results, and if the mesh is not smooth enough for discretization schemes, maintaining GCL can improve the convergence rate of numerical error by a whole order.The studies of mesh detect method is presented in the sixth chapter. To avoid the dis-sociation between mesh detect method and discretization process, the SCMM mesh detect method is proposed which correlates highly with discretization schemes, and the indicator of SCMM mesh detect method is designed to represent the aberration of a computational cell from a parallelepiped. Subsequently, the numerical tests show that the SCMM mesh detect method is sensitive to mesh smoothness and can rather represent the correlation of mesh quality with discretization process.The conclusions are presented in the seventh chapter. The major innovations of this dissertation are generalized, and the possible future works are discussed.