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ACCURACY OF SCHEMES WITH NONUNIFORM MESHES FOR COMPRESSIBLE FLUID FLOWS

IP.com Disclosure Number: IPCOM000128824D
Original Publication Date: 1985-Dec-31
Included in the Prior Art Database: 2005-Sep-19
Document File: 24 page(s) / 63K

Publishing Venue

Software Patent Institute

Related People

Eli Turkel: AUTHOR [+3]

Abstract

We consider the accuracy of the space discretization for time-dependent problems when a nonuniform mesh is used. We show that many schemes reduce to first-order accuracy while a popular finite volume scheme is even inconsistent for general grids. This accuracy is based on physical variables. However, when accuracy is measured in computational variables then second-order accuracy can be obtained. This is meaningful only if the mesh accurately reflects the properties of the solution. In addition we analyze the stability properties of some improved accurate schemes and show that they also allow for larger time steps when Runge-Kutta type methods are used to advance in time.

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THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

ACCURACY OF SCHEMES WITH NONUNIFORM MESHES FOR COMPRESSIBLE FLUID FLOWS

Eli Turkel Tel-Aviv University and Institute for Computer Applications in Science and Engineering

Abstract

We consider the accuracy of the space discretization for time-dependent problems when a nonuniform mesh is used. We show that many schemes reduce to first-order accuracy while a popular finite volume scheme is even inconsistent for general grids. This accuracy is based on physical variables. However, when accuracy is measured in computational variables then second-order accuracy can be obtained. This is meaningful only if the mesh accurately reflects the properties of the solution. In addition we analyze the stability properties of some improved accurate schemes and show that they also allow for larger time steps when Runge-Kutta type methods are used to advance in time.

Research was supported in part by the National Aeronautics and Space Administration under NASA Contract No. NASI-17070 while the author was in residence at ICASE, NASA Langley Reserach Center, Hampton, VA 23665.

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1. INTRODUCTION

With the latest class of computers it is now possible to routinely perform two-dimensional calculations for both the Euler and the compressible NavierStokes equations. Three- dimensional Euler calculations about simple shapes are feasible if a relatively coarse grid is used. Three-dimensional calculations based on the thin layer compressible Navier-Stokes calculations are just becoming available. The current trend is to use coordinates constructed so that the solid surfaces are coordinate lines while the other coordinate directions are close to orthogonal (see however [4]). This approach simplifies the boundary conditions at the solid surfaces. Moreover, whenever boundary layers exist, this approach allows one to refine the mesh across the boundary layer while having a relatively coarse mesh parallel to the boundary layer.

The construction of a general curvilinear grid is still not easily accomplished. Difficulties occur when one wishes the grids to vary smoothly while also being able to concentrate points in certain regions. Since many bodies have cusped regions, the meshes frequently are far from regular in these regions. In three dimensions these difficulties are compounded. First, on present day machines one is still restricted to relatively coarse grids. In addition the shape of the bodies can be more complicated in three dimensions. One frequently uses a quasi two- dimensional approach which has difficulties when the body no longer appears in some two- dimensional slice. The result of all these difficulties is that the meshes that are constructed are highly distorted in some regions. These distortions appear as high aspect ratios and angles quite far from 900 for quadrilateral elements. Even more disturbing is the change of these quantities from a cell to its neighboring cells (see, for exampl...