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Abstract

A fast implicit (Newton-Krylov) finite volume algorithm is developed for higher-order unstructured (cell-centered) steady-state computation of inviscid compressible flows (Euler equations). The matrix-free General Minimal Residual (GMRES) algorithm is used for solving the linear system arising from implicit discretization of the governing equations, avoiding expensive and complicated explicit computation of the higher-order Jacobian matrix. An Incomplete Lower-Upper factorization technique is employed as the preconditioning strategy and a first-order Jacobian as a preconditioning matrix. The solution process is divided into two phases: start-up and Newton iterations. In the start-up phase an approximate solution of the fluid flow is computed which includes most of the physical characteristics of the steady-state flow. A defect correction procedure is proposed for the start-up phase consisting of multiple implicit pre-iterations. At the end of the start-up phase (when the linearization of the flow field is accurate enough for steady-state solution) the solution is switched to the Newton phase, taking an infinite time step and recovering a semi-quadratic convergence rate (for most of the cases). A proper limiter implementation for higher-order discretization is discussed and a new formula for limiting the higher-order terms of the reconstruction polynomial is introduced. The issue of mesh refinement in accuracy measurement for unstructured meshes is revisited. A straightforward methodology is applied for accuracy assessment of the higher-order unstructured approach based on total pressure loss, drag measurement, and direct solution error calculation. The accuracy, fast convergence and robustness of the proposed higher-order unstructured Newton-Krylov solver for different speed regimes are demonstrated via several test cases for the 2nd, 3rd and 4th-order discretization. Solutions of different orders of accuracy are compared in detail through several investigations. The possibility of reducing the computational cost required for a given level of accuracy using high-order discretization is demonstrated.