A fast solver for the Euler equations on unstructured grids using a Newton-GMRES method

Abstract

This paper discusses the use of two variations on Newton's method, quasi-Newton and full-Newton, for the solution of the Euler equations on unstructured triangular grids. The ILU(n)-preconditioned GMRES algorithm is employed in the solution of the Jacobian matrix problem which arises at each iteration. In the quasi-Newton method, a rst-order approximation to the Jacobian matrix is used with the standard GMRES implementation. Both standard and matrix-free implementations of GM-RES are used in the full-Newton scheme, and the latter is shown to be much faster and more e cient. A hybrid scheme is presented which makes use of the strengths of both full-and quasi-Newton implementations, resulting in very fast convergence to steady state. Finally, optimal preconditioning and reordering strategies are presented.

Preconditioning methods for discontinuous Galerkin solutions of the Navier–Stokes equations

“…Matrix reordering techniques have been widely used to reduce fill in the LU factorization for direct methods used to solve large sparse linear systems [42]. These reordering techniques have also been used with ILU preconditioners of Krylov methods [11,39,10,31]. Benzi et al [10] performed numerical experiments comparing the effect of different reordering techniques on the convergence of three Krylov-subspace methods used to solve a finite difference discretization of a linear convection-diffusion problem.…”

“…Since in most aerodynamic applications the majority of the memory is used for the storage of the linearization and its factorization, such duplicate memory storage may limit the size of the problems which may be solved on a given machine [11,28,36]. In this section, an algorithm is developed that performs the incomplete-LU factorization in-place, such that no additional memory is required for the storage of the factorization.…”

“…ILU factorizations have been successfully used as preconditioners for a variety of aerodynamic problems [1,11,39,27,25,36,31]. Typically the LU factorization of a sparse matrix will have a sparsity pattern with significantly more non-zeros, or fill, than the original matrix.…”

“…The Newton-GMRES approach has been widely used for finite volume discretizations of the Euler and Navier-Stokes equations [1,12,11,39,27,25,31]. In the context of DG discretizations, GMRES was first used to solve the steady 2D compressible Navier-Stokes equations by Bassi and Rebay [8,9].…”

Parallel three‐dimensional simulation of the injection molding process

“…To reduce the bandwidth of the resulting matrices, the mesh can be first re-ordered using the reverse Cuthill-Mckee (RCM) algorithm [22]. This re-ordering strategy has been proved to increase the performance of the matrix solver by enhancing data locality [23,24].…”