Some Inaccuracies in Finite Differencing Hyperbolic Equations

Richard Grotjahn Department of Meteorology, Florida State University, Tallachassee

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James J. O'Brien Office of Naval Research, Arlington, Vd. 22217

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Abstract

The errors introduced by the use of various numerical schemes for solving mathematical models have generally been only vaguely determined previously by numerical modelers. A method for a more quantitative analysis of the inaccuracies is outlined. The error associated with some simple schemes is analyzed for several linear hyperbolic systems representative of typical problems in meteorology and oceanography. Results of previous studies of phase velocity inaccuracies are confirmed and form a basis for an extension of the analysis to group velocities. Significant angular and magnitude errors are found in the group velocity. Directional errors of 180° are found for some waves. Since the group velocity is the propagation speed of the energy, such errors may have severe consequences in a numerical model. When analysis was made of complex systems of equations, results found for simple systems reappeared. Thus, studies of simple systems may provide useful indications of behavior in more complex problems where the analysis may have to be limited. Only the long waves, i.e., those resolved by many grid points, are represented with any reasonable accuracy.

Abstract

The errors introduced by the use of various numerical schemes for solving mathematical models have generally been only vaguely determined previously by numerical modelers. A method for a more quantitative analysis of the inaccuracies is outlined. The error associated with some simple schemes is analyzed for several linear hyperbolic systems representative of typical problems in meteorology and oceanography. Results of previous studies of phase velocity inaccuracies are confirmed and form a basis for an extension of the analysis to group velocities. Significant angular and magnitude errors are found in the group velocity. Directional errors of 180° are found for some waves. Since the group velocity is the propagation speed of the energy, such errors may have severe consequences in a numerical model. When analysis was made of complex systems of equations, results found for simple systems reappeared. Thus, studies of simple systems may provide useful indications of behavior in more complex problems where the analysis may have to be limited. Only the long waves, i.e., those resolved by many grid points, are represented with any reasonable accuracy.

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