Multilevel methods for the solution of advection-dominated elliptic problems on composite grids
The author analyzes the use of multilevel algorithms in the solution of advection-dominated advection-diffusion equations in one and two dimensions. In particular, the behavior is studied of the FAC (Fast Adaptive Composite) scheme as applied to a composite grid discretization of the problem, where the discretization type is allowed to vary (from upwind to centered differences, for example) on the components of the grid. In one dimension, the analysis leads to the interpretation of a two-level version of FAC as a direct method. This analysis also provides insight into the behavior of the algorithm, and guidance for its implementation, in two dimensions. In two dimensions, the author supposes the problem to have been transformed, by an orthogonal coordinate mapping, in such a way that its characteristics are aligned with a Cartesian product grid. Various discretization strategies for this problem are studied, with emphasis placed on a finite volume method. Upwind, centered and higher-order upwind types of finite volume discretization are considered. This method is attractive in that its matrix stencils capture all of the advection of the discrete problem in their main diagonal blocks. As a result, the block Jacobi relaxation is an excellent smoother for multigrid as applied to problems on uniform subgrids. This allows formulation of highly effective multilevel algorithms for the global equations that preserve the inherent parallel nature of the strongly advection-dominated problem. The ultimate numerical method developed uses two-level FAC as an algebraic solver in a nested way to obtain an optimally efficient full multigrid-like algorithm.
- Research Organization:
- Colorado Univ., Denver, CO (United States)
- OSTI ID:
- 7297746
- Resource Relation:
- Other Information: Thesis (Ph.D.)
- Country of Publication:
- United States
- Language:
- English
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