This paper proposes a time-discontinuous and space-continuous variational integration (TDSC-VI) method for accurate elastic stress wave propagation computations in one-dimensional bar. The algorithm employs a limiter, akin to classical artificial viscosity treatment, to mitigate the deleterious Gibbs jumps across the stress discontinuities, and a parametrized consistent mass to alleviate dispersion error when the stepsizes are different from the Courant stability limit, which becomes necessary for elastic unloading and internal reflections in plastic deformation problems. Stability and accuracy analyses of the proposed TDSC-VI method are carried out and compared with several well-known traditional integration algorithms. Numerical experiments are carried out with the proposed implicit and explicit methods, which show that the proposed methods perform favorably compared to the trapezoidal rule and the central difference method. (C) 2010 Elsevier B.V. All rights reserved.