HIGH-ORDER-ACCURATE SCHEMES FOR INCOMPRESSIBLE VISCOUS FLOW

Autor: John C. Strikwerda
Rok vydání: 1997
Předmět:
Zdroj: International Journal for Numerical Methods in Fluids. 24:715-734
ISSN: 1097-0363
0271-2091
DOI: 10.1002/(sici)1097-0363(19970415)24:7<715::aid-fld513>3.0.co;2-e
Popis: SUMMARY We present new finite difference schemes for the incompressible Navier‐Stokes equations. The schemes are based on two spatial differencing methods; one is fourth-order-accurate and the other is sixth-order accurate. The temporal differencing is based on backward differencing formulae. The schemes use non-staggered grids and satisfy regularity estimates, guaranteeing smoothness of the solutions. The schemes are computationally efficient. Computational results demonstrating the accuracy are presented. 1997 by John Wiley & Sons, Ltd. High-order-accurate finite difference schemes are important in scientific computation because they offer a means to obtain accurate solutions with less work than may be required for methods of lower accuracy. Finite difference methods are attractive because of the relative ease of implementation and flexibility. In this paper we present new finite difference schemes for the incompressible Navier‐Stokes equations. The schemes are based on two spatial differencing methods, one a fourth-order-accurate method and one a sixth-order-accurate method. There are several temporal differencing methods presented in Section 7. These temporal schemes can be used with either of the spatial differencing methods. The temporal differencing is based on backward differencing formula (BDF) schemes that are used for stiff ordinary differential equations. The schemes are implicit and appear to be unconditionally stable for the Stokes equations. (A rigorous stability analysis is the subject of further research.) High-order methods have been presented by Rai and Moin 1 and Lele 2 for the fractional step method proposed by Kim and Moin, 3 There is an excellent study of these methods in the paper by Tafti. 4 A disadvantage of these methods is that because they are explicit, there is a severe stability limit on the time step. Moreover, as pointed out by Perot, 5 the pressure for fractional step methods can be no better than first-order-accurate in time. Projection methods also have difficulty with higherorder accuracy in time. 6 This is not so for the methods presented here, where the pressure can be determined to a high order of accuracy. For steady flows the method of Aubert and Deville 7 can be applied to yield fourth-order accuracy, at the expense of increasing the number of unknowns and computational complexity of the system. All these methods use staggered meshes. The schemes presented in this paper are for orthogonal Cartesian grids on non-staggered grids, the velocity components and pressure unknowns are assigned to a common grid. The schemes are for the
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