Summary

We have looked at a number of methods for solving nonlinear ordinary differential equations, including explicit, implicit, and semi-implicit numerical integration methods. Specific methods included the explicit forward Euler (similar to the finite difference approximation of §7.3.1), backward Euler (implicit), trapezoidal rule (implicit, and equivalent to the bilinear transform of §7.3.2 in the LTI case), and semi-implicit variants of the backward Euler and trapezoidal methods.

As demonstrated and discussed further in [555], implicit methods are generally more accurate than explicit methods for nonlinear systems, with semi-implicit methods (§7.4.6) typically falling somewhere in between. Semi-implicit methods therefore provide a source of improved explicit methods. See [555] and the references therein for a discussion of accuracy and stability of such schemes, as well as applied examples.

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.