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The objective of this paper is to study the numerical behavior (accuracy and numerical instability) of two high-order order single step direct integration algorithm for nonlinear dynamic. These algorithms are formulated in terms of two Hermitian finite difference operators of fifth-order local truncation error. In addition, these algorithms are unconditionally stable with no numerical damping for linear dynamic problems. The attention is devoted to the classical second-order Duffing and Van der Pol equations, as well the non-linear elastic pendulum, including the first-order Lorenz and Lotka-Volterra equations. Numerical applications compare the results including with those obtained by the second-order Newmark method

numerical methods, computation, algorithm