Improvements to a Newton-Krylov solver for aerodynamic flows

Globalized matrix-explicit Newton-GMRES for the high-order accurate solution of the Euler equations

“…A common approach [15,29] is to use the flux Jacobian of the first-order scheme for M and to use ILU decomposition to form the approximate inverse. This approach has the advantages of being easier to compute and requiring less memory than using the full-order accurate Jacobian.…”

A High-Order Accurate Unstructured Newton-Krylov Solver for Inviscid Compressible Flows

“…However, there is a restriction in increasing fill-level in practice due to memory limitation, which would affect the accuracy of preconditioning. ILU-P factorization is proven to be a robust strategy (specifically ILU-2) for GMRES preconditioning 12,15 and in general SSOR is no match for incomplete factorization even when the original matrix graph (ILU-0) has been used. 8 Our experience shows for higher-order methods using the first order preconditioner matrix, ILU-4 provides the best efficiency in preconditioning (especially in transonic flow) with the number of nonzero elements in the factorized matrix about twice the number of non-zero elements in the original preconditioner.…”

“…Newton-Krylov solvers [8][9][10][11][12][13][14] are used extensively in CFD simulations because of their property of semi-quadratic convergence when starting from a good initial solution. Since the GMRES algorithm, among other Krylov techniques, only needs matrix vector products and these products can be computed by matrix free approach, matrixfree GMRES 10 is a very practical technique for dealing with the complicated Jacobian matrices arising from higherorder discretization.…”

“…Several authors have studied the effect of various preconditioning methods on convergence of matrix-free GMRES both for structured and unstructured meshes. [8][9][10]12,15,16 Their research shows incomplete lower-upper (ILU) factorization of the approximate Jacobian is a very efficient preconditioning strategy for CFD problems. Delanaye et al 7 presented an ILU preconditioned matrixfree GMRES solver for Euler and Navier-Stokes equations on unstructured adaptive grids using quadratic reconstruction.…”

A High-Order Accurate Unstructured GMRES Algorithm for Inviscid Compressible Flows

“…Because of the non-linearity of the equations, Newton iteration would not accurately represent the solution and solving the system completely would result in excessive but not helpful computations. Wasted computation can be reduced by using Inexact-Newton method, 11 . Multiple GMRES inner iterations at each Newton outer iteration are applied to decrease the residual of the linear system by some factor American Institute of Aeronautics and Astronautics using restart.…”