Some Inaccuracies in Finite Differencing Hyperbolic Equations.

摘要

In the past, the errors introduced by use of various numerical schemes for solving mathematical models have been only vaguely determined 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 the basis for extension of the analysis to group velocities. Significant angular and magnitude errors are found in the group velocity. Directional errors of 180 deg 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.