Preconditioned Multigrid Methods for Compressible Flow Calculations on Stretched Meshes

@article{Pierce1997PreconditionedMM,
  title={Preconditioned Multigrid Methods for Compressible Flow Calculations on Stretched Meshes},
  author={Niles A. Pierce and Michael B. Giles},
  journal={Journal of Computational Physics},
  year={1997},
  volume={136},
  pages={425-445}
}
  • N. Pierce, M. Giles
  • Published 15 September 1997
  • Computer Science
  • Journal of Computational Physics
Efficient preconditioned multigrid methods are developed for both inviscid and viscous flow applications. The work is motivated by the mixed results obtained using the standard approach of scalar preconditioning and full coarsened multigrid, which performs well for Euler calculations on moderately stretched meshes but is far less effective for turbulent Naiver?Stokes calculations, when the cell stretching becomes severe. In the inviscid case, numerical studies of the preconditioned Fourier… 
Preconditioned Multigrid Methodsfor Compressible Flow Calculationson Stretched
TLDR
Determination of the analytic expressions for the preconditioned Fourier footprints in an asymptotically stretched boundary layer cell reveals that all error modes can be eeectively damped using a combination of block-Jacobi preconditionsing and a J-coarsened multigrid strategy, in which coarsening is performed only in the direction normal to the wall.
Efficient Computation of Unsteady Viscous Flows by an Implicit Preconditioned Multigrid Method
TLDR
An implicit preconditioned multlgrid algorithm is developed for the efficient solution of two-dimensional, low-frequency unsteady turbulent Navier-Stokes calculations on highly stretched meshes to yield computational savings of roughly an order of magnitude over existing methods that rely on the standard combination of scalar time stepping and full-coarsened multigrid.
Practical implementation of robust preconditioners for optimized multistage flow solvers
TLDR
A robust and practical implementation of a squared preconditioner is possible for both Euler and Navier-Stokes equations when using an analytical form in entropy variables and their corresponding transformation matrices and particular attention must be given to entropy fix and limiting techniques.
Practical Implementation and Improvement of Preconditioning Methods for Explicit Multistage Flow Solvers
TLDR
The crucial aspects of the successful implementation of a squared preconditioner which can be used in a large class of existing flow solvers that use explicit, modified Runge-Kutta methods and multigrid for convergence acceleration are exposed.
Stability Analysis of Preconditioned Approximations of the Euler Equations on Unstructured Meshes
This paper analyses the stability of a discretisation of the Euler equations on 3D unstructured grids using an edge-based data structure, first-order characteristic smoothing, a block-Jacobi
Toward Efficient Computation of Compressible and Incompressible Flows
The combination of explicit Runge-Kutta time integration with the solution of an implicit system of equations, which in earlier work demonstrated increased efficiency in computing compressible flow
Fast preconditioned multigrid solution of the Euler and Navier–Stokes equations for steady, compressible flows
TLDR
New versions of implicit algorithms are developed for the efficient solution of the Euler and Navier–Stokes equations of compressible flow, allowing solution of these problems to the level of truncation error in three to five multigrid cycles.
...
...

References

SHOWING 1-10 OF 79 REFERENCES
Efficient Computation of Unsteady Viscous Flows by an Implicit Preconditioned Multigrid Method
TLDR
An implicit preconditioned multlgrid algorithm is developed for the efficient solution of two-dimensional, low-frequency unsteady turbulent Navier-Stokes calculations on highly stretched meshes to yield computational savings of roughly an order of magnitude over existing methods that rely on the standard combination of scalar time stepping and full-coarsened multigrid.
Preconditioning compressible flow calculations on stretched meshes
TLDR
A mesh-aligned preconditioning strategy is examined which improves full coarsening multigrid performance by clustering high frequency components of the spatial Fourier footprint away from the origin for effective damping by a Runge-Kutta time stepping scheme.
Analysis of semi-implicit preconditioners for multigrid solution of the 2-D compressible Navier-Stokes equations
TLDR
Results show that the proposed preconditioners can be successfully applied to cluster residual eigenvalues for all required error modes over a wide range of governing flow parameters for most choices of discretization.
Analysis of a local matrix preconditioner for the 2-D Navier-Stokes equations
TLDR
A local matrix preconditioner based on block-Jacobi is analyzed for the linearized 2-D compressible NavierStokes equations discretized using firstand secondorder accurate upwinding to tightly cluster residual eigenvalues for high-frequency Fourier modes.
Preconditioning on stretched meshes
TLDR
A mesh-aligned preconditioning strategy is examined which is intended to improve multigrid performance in two ways: a) enhancing propagation of disturbances by shaping wave front envelopes to match cell aspect ratios, and b) clustering high frequency components of the spatial Fourier footprint away from the origin for effective damping by an optimized Runge-Kutta time stepping scheme.
Accelerating three-dimensional navier-stokes calculations
TLDR
Two alternative preconditionsed multigrid methods are proposed based on an examination of the analytic expressions for the preconditioned Fourier footprints in an asymptotically stretched boundary layer cell to dramatically accelerate convergence for three-dimensional turbulent Navier-Stokes calculations.
ANALYSIS AND DESIGN OF NUMERICAL SCHEMES FOR GAS DYNAMICS, 1: ARTIFICIAL DIFFUSION, UPWIND BIASING, LIMITERS AND THEIR EFFECT ON ACCURACY AND MULTIGRID CONVERGENCE
TLDR
A new formulation of symmetric limned positive (SLIP) schemes is presented, which can be generalized to produce schemes with arbitrary high order of accuracy in regions where the solution contains no extrema, and which can also be implemented on multi-dimensional unstructured meshes.
Numerical solution of the Euler equations by finite volume methods using Runge Kutta time stepping schemes
A new combination of a finite volume discretization in conjunction with carefully designed dissipative terms of third order, and a Runge Kutta time stepping scheme, is shown to yield an effective
The application of preconditioning in viscous flows
TLDR
A time-derivative preconditioning algorithm that is effective over a wide range of flow conditions from inviscid to very diffusive flows and from low speed to supersonic flows has been developed and convergence rates are shown to be accelerated by as much as two orders of magnitudes.
...
...