We quantify the numerical error and modeling error associated with replac- ing a nonlinear nonlocal bond-based peridynamic model with a local elasticity model or a linearized peridynamic model away from the fracture set. The non- local model treated here is characterized by a double-well potential and is a smooth version of the peridynamic model introduced in the work of Silling. The nonlinear peridynamic evolutions are shown to converge to the solution of linear elastodynamics at a rate linear with respect to the length scale $\epsilon$ of non-local interaction. This rate also holds for the convergence of solutions of the linearized peridynamic model to the solution of the local elastodynamic model. For local linear Lagrange interpolation, the consistency error for the numerical approximation is found to depend on the ratio between mesh size $h$ and $\epsilon$. More generally, for local Lagrange interpolation of order $p\geq 1$, the consistency error is of order $hp∕\epsilon$. A new stability theory for the time discretization is provided and an explicit generalization of the CFL condition on the time step and its relation to mesh size $h$ is given. Numerical simulations are provided illustrating the consistency error associated with the convergence of nonlinear and linearized peridynamics to linear elastodynamics.