Reusing preconditioners after grid refinement

Abstract

FLASH, a radiation-MHD simulation code, uses adaptive mesh refinement to concentrate computational effort near features of interest, solving a linear system at each time step using a preconditioned iterative method. After each refinement step the system is rediscretized on the new grid, rendering the old preconditioner incompatible and forcing a rebuild from scratch. This thesis investigates whether the preconditioner can instead be reused via mapping operators derived purely from the old and new grid geometries. Applied without modification, the mapped preconditioner introduces high-frequency interpolation errors that stall convergence. A composite preconditioner resolves this by surrounding the mapped application with smoothing sweeps that damp these errors. The method is first developed for a one-dimensional steady-state diffusion model problem. Extensive numerical experiments demonstrate that the composite preconditioner produces grid-independent convergence across diverse physical parameters and grid sizes. The results are robust to changes in the preconditioner, the solver, and the structure of the linear system. Applied to FLASH, a generalized transition operator handles the block-structured adaptive grid and mixed steps where refinement and derefinement occur simultaneously. An adaptive criterion triggers a rebuild only when convergence degrades past a threshold. In a 2000-step heat diffusion simulation, 1801 steps avoided a rebuild entirely, reducing wall-clock time by 4.4%. Since ILU(0) on a one-dimensional tridiagonal is among the cheapest preconditioners available, the savings are expected to grow substantially for higher-dimensional problems with more expensive builds.