Convergence

A finite-difference scheme is said to be
*convergent* if all of its solutions
in response to initial conditions and excitations converge pointwise
to the corresponding solutions of the original differential equation
as the step size(s) approach zero.

In other words, as the step-size(s) shrink, the FDS solution must improve, ultimately converging to the corresponding solution of the original differential equation at every point of the domain.

In the vibrating string example, the limit is taken as the step sizes (sampling intervals) and approach zero. Since the finite-difference approximations in Eq. (D.1) converge in the limit to the very definitions of the corresponding partial derivatives, we expect the FDS in Eq. (D.3) based on these approximations to be convergent (and it is).

In establishing convergence, it is necessary to provide that any initial conditions and boundary conditions in the finite-difference scheme converge to those of the continuous differential equation, in the limit. See [483] for a more detailed discussion of this topic.

The *Lax-Richtmyer equivalence theorem* provides a means of
showing convergence of a finite-difference scheme by showing it is
both *consistent* and *stable* (and that the initial-value
problem is *well posed*) [483]. The following
subsections give basic definitions for these terms which applicable to
our simplified scenario (linear, shift-invariant, fixed sampling
rates).

- Consistency
- Well Posed Initial-Value Problem

- Stability of a Finite-Difference Scheme
- Lax-Richtmyer equivalence theorem
- Passivity of a Finite-Difference Scheme
- Summary
- Convergence in Audio Applications

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University