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

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..

Quasi-ENO Schemes for Unstructured Meshes Based on Unlimited Data-Dependent Least-Squares Reconstruction

A crucial step in obtaining high-order accurate steady-state solutions to the Euler and NavierStokes 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 datadependent 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 opt..

An improved incremental algorithm for constructing restricted Delaunay triangulations

This paper describes one such framework, in which all problems in mesh generation and modification are treated explicitly as mesh improvement problems. This general framework can be applied to the following meshing tasks

3D flow computations over blunt bodies at hypersonic speeds using shock-fitting technique

The numerical simulation of hypersonic flows past blunt bodies by means of shock-capturing solvers is characterized by some critical drawbacks which worsen the solution quality, including stagnation point anomalies, spurious numerical oscillations, the carbuncle phenomenon, and the reduction of the order of accuracy of the solution in the entire region downstream of a captured shock. This paper describes an updated version of an unstructured shock-fitting algorithm for three-dimensional flows based on the GRUMMP (Generation and Refinement of Unstructured, Mixed-element Mesh in Parallel) library and the shock surface extraction and reconstruction process from a shock-capturing solution. The proposed technique is applied to a high speed laminar flow past a sphere, which has been extensively studied in the literature. In particular, the comparison between the shock-fitting and shock-capturing approaches clearly highlights the advantages of shock fitting, since it is able to improve the solution quality in the entire shock layer and over the body surface