<p>A high-order shock-fitting method is developed within the discontinuous flux reconstruction framework to compute shock waves accurately. The method is realized using a dynamic mesh solution approach and a robust shock detection algorithm. A strategy analogous to the method of characteristics is employed to determine the velocity of the identified shock front. The shock velocity is, then, used to move the mesh in a physically consistent manner. A one-sided operator is used to compute the common flux at the shock front, automatically satisfying the Rankine-Hugoniot condition. To accommodate the motion of the shock front, a computationally efficient mesh deformation algorithm is introduced, including a methodology that enables the shock nodes to move along the domain boundaries. In the present study, linear (Q1) and quadratic (Q2) triangular meshes are used, the latter allowing curved shocks to be represented by piecewise quadratic curves, thus achieving high-order accuracy. Several benchmark problems are used to evaluate the method’s performance and accuracy for steady applications, leading to excellent agreement with analytical solutions or data available in the literature.</p>