Title: Matrix-free second-order methods in implicit time integration for compressible flows using automatic differentiation

Authors: Frank D. Bramkamp; Bernhard Pollul; Arno Rasch; Gero Schieffer

Addresses: Chair for Computational Analysis of Technical Systems, RWTH Aachen University, D-52056 Aachen, Germany ' Chair for Numerical Mathematics, RWTH Aachen University, D-52056 Aachen, Germany ' Institute for Scientific Computing, RWTH Aachen University, D-52056 Aachen, Germany ' Chair for Computational Analysis of Technical Systems, RWTH Aachen University, D-52056 Aachen, Germany

Abstract: 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.

Keywords: matrix-free methods; Newton-Krylov; implicit time integration; automatic differentiation; QUADFLOW; ADIFOR; computational fluid dynamics; CFD; finite volume method; numerical simulation; aerodynamics; transonic flows; compressible flow; inviscid flow; airfoils; aircraft wings; viscous laminar flow; flat plates.

DOI: 10.1504/IJCSE.2014.064534

International Journal of Computational Science and Engineering, 2014 Vol.9 No.5/6, pp.484 - 498

Received: 13 Jan 2012
Accepted: 21 Sep 2012
Published online: 22 Sep 2014 *

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