Intelligent Nonlinear Solvers for Computational Fluid Dynamics

Implicit nonlinear solvers for solving systems of nonlinear PDEs are very powerful. Many compressible flow codes utilize Newton-Krylov (NK) methods and matrix-free NewtonKrylov (MFNK) methods for a range of flow regimes and different flow models such as inviscid, laminar, turbulent and reacting flows. One drawback is that these solvers are complex requiring the specification of many settings. Expertise is necessary to achieve high performance. There is a need to develop ”intelligent nonlinear solvers” that are capable of changing settings dynamically and adapting to evolving solutions and changing solver performance, in order to reduce the burden on the user, and improve overall efficiency and reliability. In this paper we take the first steps in achieving automatic control of nonlinear solvers for compressible flows by combining semi- and fully- implicit solver strategies in ways that utilizes them more efficiently than simply applying one method or another during the entire solution procedure. The understanding gained from this work will lay the groundwork for future development of more autonomous ”intelligent solvers”. Implicit solvers are widely used to in compuational fluid dynamic applications to obtain steady-state solutions to the equations governing fluid flow. Semi-implicit (point-implicit) methods are one of the most common. Semi-implicit methods are relatively easy to implement, have low memory requirements and can march at large time step sizes compared to explicit methods. Semi-implicit iterations are only modestly more expensive than explicit iterations and tend to converge linearly. They are also robust in the sense that they are relatively easy to use. However, convergence ”stalling” can be a problem in certain circumstances. In recent years, Newton-Krylov (NK) methods are becoming more popular. NK methods are less straight forward to use and more expensive per iteration than semi-implicit methods. However, NK methods are very efficient for as the solution is approached in an iterative sense, quadratic convergence rates can be achieved. Very large time steps can be used to advance the solution to steady-state. They are also robust due to the effectiveness of Krylov subspace iterative linear solvers. In both methods, solutions are achieved iteratively by solving a series of nonlinear problems where the system equations are linearized and then solved with an iterative linear solver. Semi-implicit solvers combine the nonlinear and linear loops together, solving a modified linear system less accurately but more cheaply. Typically, semi-implicit solvers are cheaper than NK methods in the beginning when the CFL is small and the linear systems are dominated by a large diagonal inertia term. Later, as the inertia term becomes smaller, the linear problem becomes more difficult to solve and NK methods become more efficient. Therefore, it would make sense to combine these approaches in a single solver stategy.