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Applicability of the Finite Element Concept to Hyperbolic Equations

Report Number: AFWAL TR 80-3048
Author(s): K. G. Guderly; Donald S. Clemm
Corporate Author: University of Dayton Research institute; Flight Dynamics Laboratory
Date of Publication: 1980-06-01
Pages: 140
Contract: AFOSR 78-3524
Project: 2304
Task: 2304N1
AD Number: ADA089774

The report analyses by means of examples the applicability of the finite element method (in the form of a weighted residual approach) to hyperbolic equations, using rectangular elements and bi-linear, bi-quadratic or bi-cubic shape functions. For sinusoidal initial conditions the errors are discussed for semi and fully discretized approximating equations. All methods have appreciable errors if the wave lengths are short. For semi-discretized methods, higher order elements give more accurate results at intermediate wave lengths. The fully discretized version for cubic elements becomes unstable, unless it is carried out as a combination of collocation and weighted residual methods. An example of a different kind shows the character of perturbations as one approaches the sonic line. A rationale for the choice of weight functions can be obtained by relating them to the Green's function. In two-dimensional problems, one can improve the cancellation of long distance effects of truncation errors by choosing characteristics as element boundaries.

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