Multigrid iterative algorithm using pseudo-compressibility for   three-dimensional mantle convection with strongly variable viscosity
  A numerical algorithm for solving mantle convection problems with strongly variable viscosity is presented. Equations for conservation of mass and momentum for highly viscous and incompressible fluids are solved iteratively by a multigrid method in combination with pseudo-compressibility and local time stepping techniques. This algorithm is suitable for large-scale three-dimensional numerical simulations, because (i) memory storage for any additional matrix is not required and (ii) vectorization and parallelization are straightforward. The present algorithm has been incorporated into a mantle convection simulation program based on the finite-volume discretization in a three-dimensional rectangular domain. Benchmark comparisons with previous two- and three-dimensional calculations including the temperature- and/or depth-dependent viscosity revealed that accurate results are successfully reproduced even for the cases with viscosity variations of several orders of magnitude. The robustness of the numerical method against viscosity variation can be significantly improved by increasing the pre- and post-smoothing calculations during the multigrid operations, and the convergence can be achieved for the global viscosity variations up to 1e10. 
Masanori Kameyama, Akira Kageyama and Tetsuya Sato