Towards Problem-Independent Multigrid Convergence Rates For Unstructured Mesh Methods I: Inviscid And Laminar Viscous Flows

This paper describes the first phase of an ongoing effort to obtain multigrid convergence rates that are independent of problem type and size. One reasonable approach for reducing physical and numerical stiffness in the context of multigrid schemes is to use local pre-conditioning to improve the distribution of the eigenvalues of the governing equations. Allmaras [2] proposed the use of block-Jacobi local pre-conditioning to restrict the eigenvalues associated with high-frequency error components to a compact region of the complex plane. If the eigenvalues are restricted in this manner, an optimal multi-stage scheme can be designed to damp these error components rapidly, which in turn should lead to good multigrid convergence. Also, to improve the convergence rate near steady-state, a Newton-GMRES scheme has been wrapped around the multigrid solver. The already-good convergence properties of the locally pre-conditioned multigrid scheme make matrix pre-conditioning for GMRES unnecessary. This allows a totally matrix-free implementation of GMRES with modest memory requirements. Results are presented for several inviscid and laminar viscous airfoil cases; results for turbulent flow will be presented in a later paper. The cases shown demonstrate that the convergence rate of the overall procedure is quite good and nearly insensitive to the type or size of problem being solved

High-Order ENO Schemes for Unstructured Meshes Based on Least-Squares Reconstruction

This article is part of an ongoing effort to develop high-order schemes for unstructured meshes to the point where meaningful information can be obtained about the trade-offs involved in using spatial discretizations of higher than second-order accuracy on unstructured meshes. This article describes a high-order accurate ENO reconstruction scheme, called DD-L 2 -ENO, for use with vertex-centered upwind flow solution algorithms on unstructured meshes. The solution of conservation equations in this context can be broken naturally into three phases: 1. Solution reconstruction, in which a polynomial approximation of the solution is obtained in each control volume. 2. Flux integration around each control volume, using an appropriate flux function and a quadrature rule with accuracy commensurate with that of the reconstruction. 3. Time evolution, which may be implicit, explicit, multigrid, or some hybrid. This article focuses primarily on solution reconstruction. A new high-order ENO reconstructio

Multigrid Acceleration of an Upwind Euler Solver on Unstructured Meshes

Multigrid acceleration has been implemented for an upwind flow solver on unstructured meshes. The flow solver is a straightforward implementation of Barth and Jespersen's unstructured scheme, with least-squares linear reconstruction and a directional implementation of Venkatakrishnan 's limiter. The multigrid scheme itself is designed to work on mesh systems which are not nested, allowing great flexibility in generating coarse meshes and in adapting fine meshes. A new scheme for automatically generating coarse unstructured meshes from fine ones is presented. A subset of the fine mesh vertices are selected for retention in the coarse mesh. The coarse mesh is generated incrementally from the fine mesh by removing one rejected vertex at a time. In this way, a valid coarse mesh triangulation is guaranteed. Factors affecting multigrid convergence rate for inviscid flow are thoroughly examined, including the effect of the number of coarse meshes used; the type of multigrid cycle employed; th..

A New Class of ENO Schemes Based on Unlimited Data-Dependent Least-Squares Reconstruction

A crucial step in obtaining high-order accurate steady-state solutions to the Euler and Navier-Stokes equations is the high-order accurate reconstruction of the solution from cell-averaged values. Only after this reconstruction has been completed can the flux integral around a control volume be accurately assessed. In this work, a new reconstruction scheme is presented that is conservative, uniformly accurate with no overshoots, easy to implement on arbitrary meshes, has good convergence properties, and is computationally efficient. The new scheme, called DD-L 2 , uses a data-dependent weighted least-squares reconstruction with a fixed stencil. The weights are chosen to strongly emphasize smooth data in the reconstruction. Because DD-L 2 is designed in the framework of k-exact reconstruction, existing techniques for implementing such reconstructions on arbitrary meshes can be used. The new scheme satisfies a relaxed version of the ENO criteria. Local accuracy of the reconstruction is o..