Matrix-free second-order methods in implicit time integration for compressible flows using automatic differentiation
F. Bramkamp, Bernhard Pollul, Arno Rasch, Gero Schieffer · International Journal of Computational Science and Engineering · 2014
In the numerical simulation of inviscid and viscous compressible fluid flow, implicit time integration schemes based on Newton-Krylov methods are frequently used. A crucial ingredient of Krylov subspace methods is the evaluation of the product of the Jacobian matrix of the spatial operator, e.g., fluxes, and a Krylov vector. In this article, we consider a matrix-free, consistently second-order accurate implementation of the Jacobian-vector product within the flow solver QUADFLOW using automatic differentiation. The convergences of the non-linear iteration using first- and second-order accurate Jacobian-vector products are compared. Different algorithmic parameters such as the CFL number and the choice of the preconditioner are taken into account. Investigations of inviscid flow around an airfoil and a wing of an aircraft, as well as the viscous laminar flow over a flat plate demonstrate that the convergence of the Newton-Krylov solver can significantly be improved using consistently second-order accurate matrix-vector products, compared to standard first-order approximations. It has been shown that a hybrid strategy using a blend between first-order and second-order accurate matrix-vector products is the best choice to obtain an efficient and robust algorithm.