Application of Newton-Krylov methodology to a three-dimensional unstructured Euler code

Abstract

A Newton-Krylov scheme is applied to an unstructured Euler code in both two and three dimensions. A simple and computationally efficient means of differencing residuals of perturbed solutions is presented that allows consistent levels of convergence to be obtained, independent of the mesh size. Results are shown for subsonic and transonic flow over an airfoil that indicate the Newton-Krylov method can be effective in accelerating convergence over a baseline scheme provided the initial conditions are sufficiently close to the root to allow the fast convergence associated with Newton's method. Two methodologies are presented to accomplish this requirement. Comparisons are made between two methods for forming the matrix-vector product used in the GMRES algorithm. These include a matrixfree finite-difference approach as well as a formulation that allows exact calculation of the matrix-vector product. The finite-difference formulation requires slightly more computer time than the exact method, but has less stringent memory requirements. Lastly, threedimensional results are shown for an isolated wing as well as for a complex-geometry helicopter configuration.