  This paper is concerned with the global stability of non-critical/critical
traveling waves with oscillations for time-delayed nonlocal dispersion
equations. We first theoretically prove that all traveling waves, especially
the critical oscillatory traveling waves, are globally stable in a certain
weighted space, where the convergence rates to the non-critical oscillatory
traveling waves are time-exponential, and the convergence to the critical
oscillatory traveling waves are time-algebraic. Both of the rates are optimal.
The approach adopted is the weighted energy method with the fundamental
solution theory for time-delayed equations. Secondly, we carry out numerical
computations in different cases, which also confirm our theoretical results.
Because of oscillations of the solutions and nonlocality of the equation, the
numerical results obtained by the regular finite difference scheme are not
stable, even worse to be blow-up. In order to overcome these obstacles, we
propose a new finite difference scheme by adding artificial viscosities to both
sides of the equation, and obtain the desired numerical results.
