GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
Journal of Computational Physics
Efficient management of parallelism in object-oriented numerical software libraries
Modern software tools for scientific computing
High-order accurate discontinuous finite element solution of the 2D Euler equations
Journal of Computational Physics
Convergence Analysis of Pseudo-Transient Continuation
SIAM Journal on Numerical Analysis
Orderings for Incomplete Factorization Preconditioning of Nonsymmetric Problems
SIAM Journal on Scientific Computing
Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems
Journal of Scientific Computing
An efficient implicit discontinuous spectral Galerkin method
Journal of Computational Physics
An assessment of linear versus nonlinear multigrid methods for unstructured mesh solvers
Journal of Computational Physics
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Jacobian-free Newton-Krylov methods: a survey of approaches and applications
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
High-order discontinuous Galerkin methods using an hp-multigrid approach
Journal of Computational Physics
A p-multigrid discontinuous Galerkin method for the Euler equations on unstructured grids
Journal of Computational Physics
SIAM Journal on Scientific Computing
Discontinuous Galerkin Methods: Theory, Computation and Applications
Discontinuous Galerkin Methods: Theory, Computation and Applications
Journal of Computational Physics
Time Implicit High-Order Discontinuous Galerkin Method with Reduced Evaluation Cost
SIAM Journal on Scientific Computing
Preconditioning for modal discontinuous Galerkin methods for unsteady 3D Navier-Stokes equations
Journal of Computational Physics
Journal of Computational Physics
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A Newton-Krylov method is developed for the solution of the steady compressible Navier-Stokes equations using a discontinuous Galerkin (DG) discretization on unstructured meshes. Steady-state solutions are obtained using a Newton-Krylov approach where the linear system at each iteration is solved using a restarted GMRES algorithm. Several different preconditioners are examined to achieve fast convergence of the GMRES algorithm. An element Line-Jacobi preconditioner is presented which solves a block-tridiagonal system along lines of maximum coupling in the flow. An incomplete block-LU factorization (Block-ILU(0)) is also presented as a preconditioner, where the factorization is performed using a reordering of elements based upon the lines of maximum coupling. This reordering is shown to be superior to standard reordering techniques (Nested Dissection, One-way Dissection, Quotient Minimum Degree, Reverse Cuthill-Mckee) especially for viscous test cases. The Block-ILU(0) factorization is performed in-place and an algorithm is presented for the application of the linearization which reduces both the memory and CPU time over the traditional dual matrix storage format. Additionally, a linear p-multigrid preconditioner is also considered, where Block-Jacobi, Line-Jacobi and Block-ILU(0) are used as smoothers. The linear multigrid preconditioner is shown to significantly improve convergence in term of number of iterations and CPU time compared to a single-level Block-Jacobi or Line-Jacobi preconditioner. Similarly the linear multigrid preconditioner with Block-ILU smoothing is shown to reduce the number of linear iterations to achieve convergence over a single-level Block-ILU(0) preconditioner, though no appreciable improvement in CPU time is shown.