A CRITICAL ANALYSIS OF THE MODIFIED EQUATION TECHNIQUE OF WARMING AND HYETT

Warming and Hyett developed a modified equation technique in which the behavior of a difference scheme is evaluated by using the coefficients of a certain modified equation. Specifically, they discovered a connection between these coefficients and the multiplication factor obtained from the von Neumann analysis. Since the dissipation and dispersion off error components are determined by the multiplication factor, the former properties can be studied using the coefficients of the modified equation. The work of Warming and Hyett represents a key step in the development of the method of modified equations. Through this work, it became clear that modified equations should be derived from the difference scheme rather than from the original differential equation. However, in order to "prove" the above connection, Warming and Hyett incorrectly interpreted their modified equations as the actual partial differential equations solved by the difference schemes. The main purpose of the current study is to investigate rigorously the above connection without using their interpretation. The result of this investigation shows that the above connection is only partially valid for multilevel schemes. In the von Neumann analysis, the multiplication factor associated with a wave number generally has (L - 1) roots for an L-level scheme. It is shown that the coefficients of the modified equation provide information for only the principal root. © 1990.