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Title: A discussion of adaptive grids and their applicability in numerical hydrodynamic modeling
Authors: University of Tennessee, Knoxville. Department of Engineering Science and Mechanics.
Mississippi State University. Department of Aerospace Engineering.
United States. Assistant Secretary of the Army (R & D)
Johnson, Billy H.
Thompson, Joe F.
Baker, A. J., 1936-
Keywords: Curvilinear coordinates
Differential equations
Numerical solutions
Numerical models
Mathematical models
Error functions
Numerical grid generation
Publisher: Hydraulics Laboratory (U.S.)
Engineer Research and Development Center (U.S.)
Series/Report no.: Miscellaneous paper (U.S. Army Engineer Waterways Experiment Station) ; no. HL-84-4.
Description: Miscellaneous Paper
Abstract: Due to the representation of a continuous physical domain in a discrete manner, errors normally occur in the numerical solution of partial differential equations. These errors are dependent upon the gradients of the solution variables as well as the spacing of the discrete set of points covering the domain. Historically, numerical errors have been controlled in one of two ways. Either a complex higher order solution scheme is employed or a fine grid spacing is constructed over the entire solution domain so that low-order schemes can be used. From a programming and computational efficiency viewpoint, low-order schemes are more desirable. However, increasing the number of computational points is often an inefficient approach since portions of the domain may contain small gradients of the solution variables and a fine grid is not required there. An approach that can sometimes be taken which reduces numerical error while still allowing for low-order schemes is the use of an adaptive grid. Such a grid moves with the developing solution so that a small grid spacing occurs in regions where the solution gradients are large with a large spacing occurring in regions where the solution is more uniform. Grid movement algorithms can be genetated either by employing a variational principle or by allowing the points to attract and/or repel each other in a manner similar to the way electrical charges behave. In either case, some measure of the error is equidistributed over the grid. Although an adaptive grid normally consists of a fixed number of points, variable node grids can also be constructed by adding and/or deleting points in particular regions so as to equidistribute the numerical error. This investigation has centered around the assessment of the usefulness of adaptive grids in the numerical solution of free surface hydrodynamics with particular problems expected to occur in the use of adaptive grids in hydrodynamic modeling being noted, e.g., the need for interpolation of the bottom topography on a moving grid. Methods of grid generation, as well as the implementation of adaptive grids, have been considered for both the finite difference and the finite element solution methods. In general, it can be concluded that adaptive grids offer significant potential for more accurate solutions at less cost when modeling the behavior of surges, hydraulic jumps, and concentration fronts, i.e., perturbation problems containing only a few high gradient regions in the computational domain at any time. However, for problems such as the propagation of short waves in coastal regions, where many wavelengths occur in the physical domain of interest at a particular time, adaptive grids offer little or no advantage since gradients occur over essentially the complete domain. Some of the greatest potential for adaptive grid techniques in hydrodynamic modeling lies in the use of such techniques to numerically generate fixed grids in tidal circulation studies. Such studies normally involve estuaries and coastal areas containing navigation channels requiring fine grid resolution. Adaptive grid techniques employing adaption to the water depth should be of great use in developing grids for such problems, and research efforts in this direction are strongly encouraged.
Rights: Approved for public release; distribution is unlimited.
Appears in Collections:Miscellaneous Paper

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