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

Abstract

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.