Abstract: A new discontinuity fitting technique based on unstructured moving grid was proposed in this paper, which contained these processes: (1) locating the initial position of discontinuities on the basis of computed flow-field by capturing algorithm; (2) dealing with the motions and deformations of fitted discontinuities using moving grid approach and remesh technique in the context of arbitrary Lagrange-Euler (ALE) formulation, at the same time ensuring to satisfy discrete geometric conservation law (DGCL); (3) ensuring the time accuracy in the process of interpolating among meshes by a high accurate transferring flow information method. A Sod problem was chosen to introduce the principle of new discontinuity fitting technique.