Alternative Solution Algorithms For Primal And Adjoint Incompressible Navier-Stokes

Regardless of the specific discretisation framework, the discrete incompressible Navier-Stokes equations present themselves in the form of a non-linear, saddle-point Oseentype system. Traditional CFD codes typically solve the system via the well-known SIMPLE-like algorithms, which are essentially block preconditioners based on Schur complement theory. Due to their “segregated” nature, which reduces to iteratively solving a sequence of linear systems smaller than the full Oseen and better conditioned, traditional SIMPLE-like algorithms
have long been considered as the only viable strategy. 

However, recent progress in computational power and linear solver capabilities has led researchers to develop, for Oseen-type systems (and discrete Navier-Stokes in particular), a number of alternative preconditioners and solution schemes, found to be more efficient than SIMPLE-like strategies but previously deemed practically unfeasible in industrial contexts.

The improved efficiency of novel preconditioners entails a) faster, more stable convergence and b) the possibility of driving residuals below more strict tolerances, which is sometimes difficult with SIMPLE due to stagnating behaviour. The second aspect in particular is extremely relevant in the context of adjoint-based optimisation, as evidence suggests that an adjoint system may be affected by convergence issues when the primal flow solution is not well converged. 

In this work, we present some solution schemes (both traditional and novel) implemented for the Mixed Hybrid Finite Volumes Navier-Stokes solver we introduced in our previous work. Performance, in terms of robustness and convergence properties, is assessed on a series of benchmark test cases. We also turn our attention to the discrete adjoint Navier-Stokes problem itself, which in essence requires solving a linear system similar to the original Oseen and  therefore may benefit from the same preconditioning techniques. We show how the primal algorithms are adapted to the adjoint system, and we run a series of adjoint test cases to compare performance of various solution schemes.