Practical implementation of robust preconditioners for optimized multistage flow solvers

Explicit multistage multigrid methods have carved their niche in the solution of complex inviscid and viscous flows. Because of their scalable parallel implementations, they have been a popular choice for the solution of large-scale, complex configuration problems. The convergence rate of these methods deteriorates when they face problems such as numerical stiffness or directional decoupling that result from propagative disparity, cell stretching, and flow alignment. Moreover, in the limit of low Mach numbers, most compressible flow solvers become ill-conditioned in addition to losing accuracy. It is possible to improve existing codes that typically use local time-stepping by using squared preconditioning schemes that combine Block-Jacobi preconditioning with low-Mach preconditioning. A robust and practical implementation of a squared preconditioner is possible for both Euler and Navier-Stokes equations when using an analytical form in entropy variables and their corresponding transformation matrices. Particular attention must be given to entropy fix and limiting techniques. Results of this squared preconditioned approach are presented for both two- and three-dimensional test cases. For these test cases the preconditioning produces convergence acceleration for all Mach numbers while always maintaining the accuracy of the solution. Additional convergence acceleration can be obtained via the optimization of various input parameters such as multistage coefficients, artificial dissipation coefficients and the CFL numbers.