Homotopy continuation of the high-order flux reconstruction/correction Procedure via reconstruction (FR/CPR) method for steady flow simulation

A homotopy continuation method is developed for the high-order flux reconstruction/correction procedure via reconstruction (FR/CPR) framework to solve hyperbolic conservation laws at steady states. The efficacy of homotopy continuation has been demonstrated using both one and two dimensional steady flow problems. For transonic/supersonic flows, the localized Laplacian artificial viscosity technique is used to stabilize high-order accurate schemes. With this technique, homotopy continuation shows good convergence properties for stiff steady transonic/supersonic shock-capturing problems. Besides, the developed homotopy continuation strategy for the high-order FR/CPR method can work robustly on steady shock capturing for a wide range of grid sizes and polynomial orders without parameter tuning in the localized Laplacian artificial viscosity.