The interaction between the solar wind and the interstellar medium is modeled self-consistently using numerical solutions of the time-dependent gasdynamic equations in spherical and cylindrical coordinates. For the results presented here it is assumed that the solar system moves through the surrounding medium with a supersonic velocity. After an initial (nonequilibrium) state has been specified, the numerical solution follows the evolution in time until the interaction relaxes to a dynamic equilibrium. As would be expected, the solutions show the formation of a bow shock upstream of the traveling solar system to deflect the interstellar plasma around the cavity created by the solar wind. A termination shock also forms to slow and compress the solar wind plasma. For the simulation in spherical coordinates, the downstream portion of the termination shock reaches equilibrium more than three times further from the Sun than the equilibrium distance to the termination shock on the upstream side.