A robust monolithic nonlinear Newton method for the compressible Reynolds averaged Navier–Stokes Equations

The goal of this work is the development of a fully monolithic nonlinear Newton algorithm for the unstructured vertex-based finite volume algorithm presented in [Akkurt and Sahin, An efficient edge based data structure for the compressible RANS equations on hybrid unstructured meshes. International Journal for Numerical Methods in Fluids, 94:13-31, (2022)]. A special attention is paid for the highly accurate construction of the first- and second-order Jacobian matrices for the Navier–Stokes equations and the one-equation negative Spalart–Allmaras turbulence model. The inviscid flux Jacobian matrices are evaluated exactly using the source code transformations provided by the INRIA Tapenade library. The implementation of the nonlinear Newton method is carried out using the PETSc Scalable Nonlinear Equations Solvers (SNES) with a line search technique. The method can utilize both the Jacobian-free finite difference and direct approaches for the Jacobian vector product. The INRIA pyAMG anisotropic mesh adaptation library has been integrated in order to further improve its numerical accuracy at a lower computational cost. The algorithm has been applied to the two- and three-dimensional mesh convergence test cases from the 4th AIAA CFD High Lift Prediction Workshop in order to demonstrate its robustness in achieving machine precision for a realistic high-lift system. The highly accurate Jacobian evaluation process and high CFL numbers are found to be very critical for achieving machine precision. The mesh adaptation is also proven to be very robust in refining regions with high gradients, which is especially important for the simulations at high angles of attack, where it is rather difficult to determine refinement regions a priori. The anisotropic mesh adaptation studies are carried out for up to 92,993,470 vertices in three-dimensions, and the numerical results indicate very good agreement with the committee's experimental data.