The paper describes the application of front tracking to the polymer system, an example of a nonstrictly hyperbolic system. Front tracking computes piecewise constant approximations based on approximate Riemann solutions and exact tracking of waves. It is well known that the front tracking method may introduce a blowup of the initial total variation for initial data along the curve where the two eigenvalues of the hyperbolic system are identical. It is demonstrated by numerical examples that the method converges to the correct solution after a finite time, and that this time decreases with the discretization parameter. For multidimensional problems, front tracking is combined with dimensional splitting, and numerical experiments indicate that large splitting steps can be used without loss of accuracy. Typical CFL numbers are in the range 10-20, and comparisons with Riemann free, high-resolution methods confirm the high efficiency of front tracking. The polymer system, coupled with an elliptic pressure equation, models two-phase, three-component polymer flooding in an oil reservoir. Two examples are presented, where this model is solved by a sequential time stepping procedure. Because of the approximate Riemann solver, the method is non-conservative and CFL numbers must be chosen only moderately larger than unity to avoid substantial material balance errors generated in near-well regions after water breakthrough. Moreover, it is demonstrated that dimensional splitting may introduce severe grid orientation effects for unstable displacements that are accentuated for decreasing discretization parameters.