  This paper considers the Cauchy problem for the nonlinear dynamic string
equation of Kirchhoff-type with time-varying coefficients. The objective of
this work is to develop a time domain discretization algorithm capable of
approximating a solution to this initial-boundary value problem. To this end, a
symmetric three-layer semi-discrete scheme is employed with respect to the
temporal variable, wherein the value of a nonlinear term is evaluated at the
middle node point. This approach enables the numerical solutions per temporal
step to be obtained by inverting the linear operators, yielding a system of
second-order linear ordinary differential equations. Local convergence of the
proposed scheme is established, and it achieves quadratic convergence regarding
the step size of the discretization of time on the local temporal interval. We
have conducted several numerical experiments using the proposed algorithm for
various test problems to validate its performance. It can be said that the
obtained numerical results are in accordance with the theoretical findings.
