We analyze a pair of nonlinear PDEs describing viscoelastic fluid flow in one dimension. We give a summary of the physical derivation and nondimensionlize the PDE system. Based on the boundary conditions and parameters, we are able to classify three different categories of traveling wave solutions, consistent with the results in [?]. We extend this work by analyzing the stability of the traveling waves. We thoroughly describe the numerical schemes and software program, VISCO, that were designed specifically to analyze the model we study in this paper. Our simulations lead us to conjecture that the traveling wave solutions found in [?] are globally stable for all sets of initial conditions with the appropriate asymptotic boundary conditions. We are able give some analytical evidence in support of this hypothesis but are unsuccessful in providing a complete proof.