National Aeronautics and

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National Aeronautics and (2001)

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by Space Administration Langley , Dimitri J. Mavriplis , Dimitri J. Mavriplis

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J. Comput. Phys

BibTeX

@ARTICLE{Langley01nationalaeronautics,
    author = {Space Administration Langley and Dimitri J. Mavriplis and Dimitri J. Mavriplis},
    title = {National Aeronautics and},
    journal = {J. Comput. Phys},
    year = {2001},
    volume = {175},
    pages = {302--325}
}

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Abstract

The relative performance of a non-linear FAS multigrid algorithm and an equivalent linear multigrid algorithm for solving two di#erent non-linear problems is investigated. The #rst case consists of a transient radiation-di#usion problem for which an exact linearization is available, while the second problem involves the solution of the steady-state Navier-Stokes equations, where a #rst-order discrete Jacobian is employed as an approximation to the Jacobian of a second-order accurate discretization. When an exact linearization is employed, the linear and non-linear multigrid methods converge at identical rates, asymptotically, and the linear method is found to be more e#cient due to its lower cost per cycle. When an approximate linearization is employed, as in the Navier-Stokes cases, the relative e#ciency of the linear approachversus the non-linear approach depends both on the degree to which the linear system approximates the full Jacobian as well as the relative cost of linear versus non-linear multigrid cycles. For cases where convergence is limited by a poor Jacobian approximation, substantial speedup can be obtained using either multigrid method as a preconditioner to a Newton-Krylov method.

Keyphrases

national aeronautics exact linearization linear method steady-state navier-stokes equation relative performance substantial speedup poor jacobian approximation equivalent linear multigrid algorithm non-linear fa multigrid algorithm rst-order discrete jacobian di erent non-linear problem transient radiation-di usion problem identical rate linear system non-linear approach relative ciency non-linear multigrid cycle second-order accurate discretization approximate linearization navier-stokes case second problem non-linear multigrid method converge multigrid method rst case full jacobian newton-krylov method relative cost

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