A Discontinuous High-Order Shock-Fitting Method for Complex Flow Problems

The present work extends a high-order shock-fitting algorithm to handle unsteady and viscous flow problems. Although the formulation is developed within the flux reconstruction framework, it can be readily adapted to other discontinuous spectral element-type methods. The proposed approach comprises three main components. The first is a simple yet versatile discontinuity detection algorithm that employs high-order and cell-averaged solution jumps to accurately identify cell faces containing a discontinuity. The second is a shock front motion formulation that determines the shock velocity at a given time and moves the nodes associated with the shock front accordingly. The third is a mesh deformation algorithm that modifies the entire mesh to prevent warping or the occurrence of negative Jacobians. In cells identified as containing a discontinuity, the Rankine–Hugoniot conditions are enforced implicitly using an upwind numerical flux. The present study addresses challenging two-dimensional steady-state and transient cases, both inviscid and viscous, including curved discontinuities and shock interactions. These problems are discretized using quadratic (Q2) triangular meshes, enabling accurate representation of shock curvature within the mesh. The equations are solved using schemes of first- to fourth-order spatial accuracy, combined with both implicit and explicit time-marching strategies. The shock-fitting approach demonstrates strong performance across all benchmark cases considered.