Search NASA⌕ Search

SEARCH · Search NASA

Results for “Stage-Value predictors”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

3 records
All sourcesNASA NTRSDOE OSTI
NASA NTRS2024

Intrastep, Stage-Value Predictors for Diagonally-Implicit Runge-Kutta Methods

To better identify the necessary attributes of good stage-value predictors (SVPs), numerous SVPs are designed for an existing: ESDIRK4(3)7L[2]SA [26] and a new: ESDIRK4(3)8L[2]SA scheme.1 Both are stifflyaccurate, stage-order two, explicit, singly-diagonally implicit Runge–Kutta (ESDIRK) schemes. Tradeoffs are studied in the parameter spaces enforcing the constraints on accuracy, linear stability, nonlinear stability and coefficient size to determine which objectives correlate with effective predictors. The SVPs are tested on three challenging external aerodynamics problems [107 − 108 degrees of freedom (DoFs)], each with a different level of stiffness. The problems include two 3D airfoils simulations and one canonical turbulence simulation. All simulations use the compressible Navier-Stokes equations (CNSE). An entropy stable spectral collocation formulation is used for discretizing the spatial terms in the equations. Simulations are performed at a wide variety of temporal error tolerances. Problems that are sufficiently stiff (e.g., lax temporal error tolerances) benefit from SVPs designed with second-order accuracy and stability properties: A-stability, and L-stability, rather than high accuracy constraints. Simulations with modest stiffness (e.g., strict error tolerances) are better suited for SVPs designed using high accuracy constraints. Designing SVPs with enhanced stability properties is tedious but worthwhile. Simulation times are reduced with optimal SVPs by as much as 100% on some stages, with combined stepwise improvements of between 50 − 100% for both methods. A comparative study is performed with the two aforementioned methods as well as four other ESDIRKs. The newly designed ESDIRK4(3)8L[2]SA with γ ≈ 1/10, proves to be the most efficient of the six tested ESDIRK schemes simulating the CNSE.

Diagonally-Implicit Runge-Kutta↗
NASA NTRS2001

Additive Runge-Kutta Schemes for Convection-Diffusion-Reaction Equations

Additive Runge-Kutta (ARK) methods are investigated for application to the spatially discretized one-dimensional convection-diffusion-reaction (CDR) equations. First, accuracy, stability, conservation, and dense output are considered for the general case when N different Runge-Kutta methods are grouped into a single composite method. Then, implicit-explicit, N = 2, additive Runge-Kutta ARK2 methods from third- to fifth-order are presented that allow for integration of stiff terms by an L-stable, stiffly-accurate explicit, singly diagonally implicit Runge-Kutta (ESDIRK) method while the nonstiff terms are integrated with a traditional explicit Runge-Kutta method (ERK). Coupling error terms are of equal order to those of the elemental methods. Derived ARK2 methods have vanishing stability functions for very large values of the stiff scaled eigenvalue, z(exp [I]) goes to infinity, and retain high stability efficiency in the absence of stiffness, z(exp [I]) goes to zero. Extrapolation-type stage-value predictors are provided based on dense-output formulae. Optimized methods minimize both leading order ARK2 error terms and Butcher coefficient magnitudes as well as maximize conservation properties. Numerical tests of the new schemes on a CDR problem show negligible stiffness leakage and near classical order convergence rates. However, tests on three simple singular-perturbation problems reveal generally predictable order reduction. Error control is best managed with a PID-controller. While results for the fifth-order method are disappointing, both the new third- and fourth-order methods are at least as efficient as existing ARK2 methods while offering error control and stage-value predictors.

Kennedy, Christopher A.↗