دانلود رایگان مقاله درون یابی چند جمله ای با برون یابی ریچاردسون برای کاهش خطای گسسته در CFD

ABSTRACT

 The goal of the present study is to present and test a novel numerical procedure for reducing the discretization error associated with several types of variables of interest in basic Computational Fluid Dynamics (CFD) problems. Variables of interest are classified into five types according to their locations on various grids. According to the current literature, Repeated Richardson Extrapolation (RRE) performs well for only one of the five types of variable, i.e., for global variables or those that otherwise have fixed nodal positions on different grids. RRE does not perform well for the remaining four variable types. Because of this limitation, in this work, polynomial interpolation is applied to various numerical solutions obtained on different grids, followed by RRE. Four problems are used to test the proposed procedure, one linear and three non-linear based on the following equations: 1D Poisson, 2D Burgers and 2D Navier–Stokes. These equations are discretized using the Finite Difference method with approximations of second- and fourth-order accuracy and the Finite Volume method with approximations of first- and second-order accuracy. Polynomial interpolation functions for one- and two-dimensional domains are adopted, and optimization techniques are also adopted in some cases. The discretization error is significantly reduced, and the order of accuracy is also increased: for example, based on a second-order scheme with an error of 1.4×10−6, we obtain 2.1×10−27 using six extrapolations on a grid with 1460 elements and an order of accuracy of 14.5. The computational effort (CPU time and memory usage) needed to obtain the solution at a given level of numerical error or using a specific grid is also significantly reduced.

4. Conclusion

Based on previously published reports of the poor performance of RRE in reducing Eh for some types of variables of interest, five types of variables were defined in this work according to their locations on different grids. The performance of RRE when applied to these variables was then analyzed using four test problems (1D Poisson equation, 2D Burgers equations and the 2D Navier–Stokes equations), discretized using Finite Difference and Finite Volume Methods, including numerical schemes of various orders of accuracy. Grids with as many as millions of nodes, several levels of extrapolation, multigrid methods and quadruple and double precision were used. Based on the obtained results, it was concluded that to employ RRE, the variable of interest must firstly be characterized as one of the five established types. The subsequent procedure for the reduction of Eh using RRE is as follows: (1) For a global or local variable with a fixed location that coincides with a nodal point on every considered grid, RRE can be directly applied. RRE is best suited to this type of variable. (2) For a variable located at the midpoint of a set of nodal coordinates, linear (or bilinear/trilinear) interpolation should be applied prior to RRE. (3) For a variable that has a fixed location that is neither a nodal point nor a midpoint thereof, polynomial interpolation of the highest possible degree appropriate to the number of dimensions of the problem under study must be applied prior to RRE. (4) For a variable whose value is predetermined and whose position is unknown, polynomial interpolation of the highest possible degree should be applied before finding the solution of the associated polynomial equation; only then RRE can be applied. (5) In the case of a variable that involves finding an extremum, in which the coordinate values may change as the grid is refined, polynomial interpolation of the highest possible degree appropriate to the number of dimensions of the problem under study should be applied, followed by an optimization technique before applying RRE. The procedure proposed in the present article permits the reduction of the numerical error of RRE in those cases in which RRE is considered to be ineffective according to the current literature. This procedure, which is characterized as a post-processing procedure, entails extremely low computational cost. Given a desired value for the numerical error, the CPU time and RAM memory required to achieve it are much lower when this procedure is used than when it is not. Moreover, it is, in principle, valid for schemes of any order of accuracy that use either Finite Difference or Finite Volume methods and for any equation and number of dimensions, and it permits considerable reduction in the numerical error.