Other well known numerical integration methods for ODEs include
second-order backward difference formulas (commonly used in circuit
simulation [556]), the fourth-order Runge-Kutta method
[99], and their various explicit, implicit, and semi-implicit
variations. See [556] 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 [556]
include electric-guitar distortion devices
[555,557], the classic ``tone stack''
[554] (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
[556] 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
[558,341,295,84,266,367,400].

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.

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