Steepest descent optimisation of Runge-Kutta coefficients for second order implicit finite volume CFD codes

One of the key research topics in the computational fluid dynamics community is to improve the computational efficiency of steady-state finite volume codes. Real-world use cases require the solution to the Navier-Stokes equations for a wide range of Mach numbers, Reynolds numbers and mesh cell aspect ratios. This introduces stiffness in the discretised equations and therefore a slowdown in convergence. The community has pursued in particular two avenues to speed up the convergence of the corresponding error modes: Optimisation of Runge-Kutta coefficients for explicit Runge-Kutta schemes; and the introduction of implicit preconditioners, with a limited investigation of Runge-Kutta coefficients suitable to those implicit preconditioners. After proposing improvements to the implicit preconditioner, the present work proposes an optimisation procedure allowing the optimisation of the Runge-Kutta coefficients specifically for the implicit preconditioner. Employed on a realistic use case, the Runge-Kutta coefficients extracted with this method show a 20%-38% reduction of the number of iterations needed for convergence compared to Runge-Kutta coefficients recommended in the literature for comparable schemes and with the same computational cost per iteration.