A Jacobian-free Edge-based Galerkin Formulation for Compressible Flows

•LU-SGS has been improved from the version originally proposed by Luo.•The JFNK strategy can address the MUSCL reconstructed and slope-limited states.•The JFNK strategy significantly reduces the total memory footprint.•The performance of the JFNK strategy has been assessed extensively. A parallel formulation of a Jacobian-free all Mach numbers solver on unstructured hybrid meshes is proposed. The Finite Element formulation is edge-based with flow stabilization obtained with either AUSM+-up or Roe scheme. The linear system is solved via a Jacobian-Free Newton-Krylov (JFNK) method with Lower-Upper Symmetric Gauss-Seidel (LU-SGS) used as matrix-free preconditioner. The traditional formulation of LU-SGS is enriched by including the contributions from viscous fluxes and boundary conditions. The accuracy and efficiency of the proposed approach are demonstrated over cases ranging from low to high Mach numbers: subsonic flow over the Trap Wing, transonic flow over the ONERA M6 wing, supersonic flow over a sphere, supersonic flow over a waverider and finally hypersonic flow over a sphere.