An accurate and efficient numerical method for steady, two-dimensional Euler equations is applied to study steady shock waves perpendicular to smooth, convex surfaces. The main subject of study is the flow near both ends of the shock wave: the shock-foot and shock-tip flow. A known analytical model of the inviscid shock-foot flow is critically investigated, analytically and numerically. The results obtained agree with those of the existing analytical model. For the inviscid shock-tip flow, two existing analytical solutions are reviewed. Numerical results are presented which agree with one of these two solutions. Good numerical accuracy is achieved through a monotone, second-order accurate, finite-volume discretization. Good computational efficiency is obtained through iterative defect correction iteration and a multigrid acceleration technique which employs local grid refinement.