Further Reading in Nonlinear Methods

Other well known numerical integration methods for ODEs include second-order backward difference formulas (commonly used in circuit simulation [558]), the fourth-order Runge-Kutta method [99], and their various explicit, implicit, and semi-implicit variations. See [558] 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 [558] include electric-guitar distortion devices [557,559], the classic ``tone stack'' [556] (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 [558] 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 [560,341,295,84,266,367,400].

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.