September 12,2001 we propose a fully conservative front tracking algorithm in two space dimension. This algorithm first uses the point shifted algorithm on two adjacent time levels and then constructs space time hexahedra as computational units. We develop and prove a successful geometric construction under certain interface requirement. The algorithm has a first order local truncation error for cells near the tracked discontinuity, which is an improvement by one order of accuracy over most finite difference schemes, which have O (1) local truncation errors near discontinuities. September 12, 2001 we propose a fully conservative front tracking algorithm in two space dimension. This algorithm first uses the point shifted algorithm on two adjacent time levels and then constructs space time hexahedra as computational units. We develop and prove a successful geometric construction under interface requirement. The algorithm has a first order local truncation error for cells near the tracked discontinuity, which is an improvement by one order of accuracy over most finite difference schemes, which have O (1) local truncation errors near discontinuities.