Application of a Jacobian-Free Newton-Krylov Method to the Simulation of Hypersonic Flows

Many production-level hypersonic flow codes employ a defect-correction approach to solving the partial differential equations used to model high-speed thermochemical nonequilibrium flows. Although this approach typically provides fast convergence during the startup phase of a simulation, it is limited to a linear asymptotic convergence rate. Newton-Krylov methods are an alternative approach that can achieve a quadratic asymptotic convergence rate if an exact Jacobian is used in the implicit operator. The challenge is that the derivation of an exact linearization of a hypersonic flow code is non-trivial and prone to human error. In this paper, we present our work towards overcoming this challenge via the Jacobian-Free Newton-Krylov method. The method is Jacobian-free in the sense that the algorithm only requires matrix-vector products which are obtained by Fr\'{e}chet derivatives. In our solver, we employ complex-step Fr\'{e}chet derivatives. By doing so the accuracy of our numerical derivatives are not restricted by subtractive floating-point error, and as such we are able to get a Jacobian influence that is effectively as accurate as one that is analytically derived and evaluated. We demonstrate the solver's performance on several test cases that are representative of the type of flows of interest to the hypersonic community. The convergence rate of the developed Newton-Krylov solver is compared against a point-implicit implementation based on a defect-correction method.