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.