Research Summaries

Back Multi-Rate High-Order Time-Integrators for Adaptive Local High-Order Discretization Methods

Fiscal Year 2014
Division Graduate School of Engineering & Applied Science
Department Applied Mathematics
Investigator(s) Giraldo, Francis X.
Sponsor Air Force Office of Scientific Research (Air Force)
Summary We propose to develop high-order accurate and highly efficient time-integration methods for local high-order spatial discretizations of hyperbolic partial differential equations (PDEs) of interest to the Air Force; we will use both spectral elements and discontinuous Galerkin methods as representative of the state-of-the-art local high-order methods and the compressible Navier-Stokes equations as representative of the system of PDEs. In a previous AFOSR-funded project, the PIs developed a general implicit-explicit (IMEX) time-integration framework that allows any IMEX method to be implemented into the same computer code in a unified way. We showed how to do this for multi-step methods as well as single-step multi-stage methods; in addition, we developed a new second-order IMEX Runge-Kutta method (Giraldo, 2003. In that project, we studied both explicit and IMEX methods and explored fully-implicit methods (using single-step multi-stage methods) where we showed that using adaptive time-stepping in conjunction with dense output formulas allowed the fully-implicit methods to be more competitive (in terms of efficiency) with IMEX methods for low Mach number simulations; we propose to continue this work to extend the approach to grid-adaptive simulations. All of the work done in the previous AFOSR-funded proposal assumed that all the grid cells in the computational domain use a single (global) time-step to evolve the solutions forward in time. These are known as single-rate methods. However, one of the goals of the proposed work is to develop efficient time-integration strategies for grid-adaptive simulations, i.e., simulations whereby the grid changes dynamically with features of the flow deemed to be important (Kopera and Giraldo JCP 2013}. We propose to extend our work on explicit and IMEX methods to include multi-rate methods; these are methods that allow each grid cell to use a different time-step. The idea is to let the small grid cells use small time-steps while the large grid cells use large time-steps while maintaining the formal order of accuracy of the method and remaining synchronized in time. Finally, since all implicit time-integrators require the use of iterative solvers and preconditioners, we propose to compare and contrast various iterative solvers (including GMRES, BiCGStab, and Richardson methods) to see which method yields better performance on massively parallel computers (we will consider core counts at least in the tens of thousands). To make our iterative methods converge faster, we will extend our recent work on preconditioners (Carr et al SISC 2012) to allow faster convergence under adaptive mesh refinement via more efficient preconditioner recomputation whenever the grid changes (Carr et al in preparation 2013).
Publications Publications, theses (not shown) and data repositories will be added to the portal record when information is available in FAIRS and brought back to the portal
Data Publications, theses (not shown) and data repositories will be added to the portal record when information is available in FAIRS and brought back to the portal