The mixed convolved action principle provides a new rigorous weak variational formalism for a broad range of initial boundary value problems in mathematical physics and mechanics in terms of mixed formulation, convolution, and fractional calculus. In this paper, its potential in the development of numerical methods for transient problems in various dynamical systems when adopting temporally second order approximation is investigated. For this, the classical single-degree-of-freedom linear elastic dynamical systems are primarily considered to investigate computational characteristics of the developed algorithms. For the undamped system, all the developed algorithms are symplectic with respect to the time step. For the damped system, they are shown to be accurate with good convergence characteristics.