Assessment of Linear Versus Non-linear Multigrid Methods for Unstructured Mesh Solvers

摘要

The relative performance of a non-linear FAS multigrid algorithm and an equivalent linear multigrid algorithm for solving two different non-linear problems is investigated. The first case consists of a transient radiation- diffusion problem for which an exact linearization is available, while the second problem involves the solution of the steady-state Navier-Stokes equations, where a first-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 efficient due to its lower cost per cycle. When an approximate linearization is employed, as in the Navier-Stokes cases, the relative efficiency of the linear approach versus 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.