This paper describes and compares the zeroth-order semi-discretization, spectral element, and Legendre collocation methods. Each method is a technique for solving delay differential equations (DDEs) as well as determining regions of stability in the DDE parameter space. We present the necessary concepts, assumptions, and equations required to implement each method. To compare the relative performance between the methods, the convergence rate achieved and computing time required by each method are determined in two numerical studies consisting of a ship stability example and the delayed damped Mathieu equation. For each study, we present a stability diagram in parameter space and a convergence plot. The spectral element method is demonstrated to have the quickest convergence rate while the Legendre collocation method requires the least computing time. The zeroth-order semi-discretization method on the other hand has both the slowest convergence rate and requires the most computing time.

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