Showing papers by "William J. Rider published in 2003"

Combined space and time convergence analysis of a compressible flow algorithm

Abstract: In this study, we quantify both the spatial and temporal convergence behavior simultaneously for various algorithms for the two-dimensional Euler equations of gasdynamics. Such an analysis falls under the rubric of verification, which is the process of determining whether a simulation code accurately represents the code developers description of the model (e.g., equations, boundary conditions, etc.). The recognition that verification analysis is a necessary and valuable activity continues to increase among computational fluid dynamics practicioners. Using computed results and a known solution, one can estimate the effective convergence rates of a specific software implementation of a given algorithm and gauge those results relative to the design properties of the algorithm. In the aerodynamics community, such analyses are typically performed to evaluate the performance of spatial integrators; analogous convergence analysis for temporal integrators can also be performed. Our approach combines these two usually separate activities into the same analysis framework. To accomplish this task, we outline a procedure in which a known solution together with a set of computed results, obtained for a number of different spatial and temporal discretizations, are employed to determine the complete convergence properties of the combined spatio-temporal algorithm. Such an approach is of particular interest formore » Lax-Wendroff-type integration schemes, where the specific impact of either the spatial or temporal integrators alone cannot be easily deconvolved from computed results. Unlike the more common spatial convergence analysis, the combined spatial and temporal analysis leads to a set of nonlinear equations that must be solved numerically. The unknowns in this set of equations are various parameters, including the asymptotic convergence rates, that quantify the basic performance of the software implementation of the algorithm.« less

Adaptive Time Integration for Hyperbolic Conservation Equations

Abstract: We introduce a fundamentally new time integration method for hyperbolic conservation laws based on self-adaptivity of the temporal method itself. The adaptivity is based upon the smoothness of the solution measured locally in time. Our approach can be contrasted with the usual global selection of a time integration methods and error-based time step selection methodology. A challenge to this approach is maintaining the adaptivity and the conservation form. These methods are challenged with several standard problems as well as high-resolution experimental data of shock-driven mixing (Richtmyer-Meshkov).