Further Reading in Nonlinear Methods

Other well known numerical integration methods for ODEs include second-order backward difference formulas (commonly used in circuit simulation [555]), the fourth-order Runge-Kutta method [99], and their various explicit, implicit, and semi-implicit variations. See [555] for further discussion of these and related finite-difference schemes, and for application examples in the virtual analog area (digitization of musically useful analog circuits). Specific digitization problems addressed in [555] include electric-guitar distortion devices [553,556], the classic ``tone stack'' [552] (an often-used bass, midrange, and treble control circuit in guitar amplifiers), the Moog VCF, and other electronic components of amplifiers and effects. Also discussed in [555] is the ``K Method'' for nonlinear system digitization, with comparison to nonlinear wave digital filters (see Appendix F for an introduction to linear wave digital filters).

The topic of real-time finite difference schemes for virtual analog systems remains a lively research topic [554,338,293,84,264,364,397].

For Partial Differential Equations (PDEs), in which spatial derivatives are mixed with time derivatives, the finite-difference approach remains fundamental. An introduction and summary for the LTI case appear in Appendix D. See [53] for a detailed development of finite difference schemes for solving PDEs, both linear and nonlinear, applied to digital sound synthesis. Physical systems considered in [53] include bars, stiff strings, bow coupling, hammers and mallets, coupled strings and bars, nonlinear strings and plates, and acoustic tubes (voice, wind instruments). In addition to numerous finite-difference schemes, there are chapters on finite-element methods and spectral methods.

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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.