A strategy for choosing Gegenbauer reconstruction parameters for numerical stability

The Gegenbauer reconstruction method was first proposed in 1992, but in early studies no attempts were made to optimize the relevant parameters of this method. These parameters were allowed to grow proportionally with the number of nodes which, in many cases, resulted in exponential convergence for a selected range of the proportionality constants. Early studies also made clear that very large error bounds could be expected if these key parameters were not chosen carefully. Subsequent studies then pointed out that, although unrelated to the method's analytically predictable domains of poor accuracy, round-off errors could also sabotage the method's accuracy. The challenge of successfully implementing a Gegenbauer reconstruction then rests on understanding the performance trade-offs we can expect when choosing the key parameters in accordance with different objectives. In this study, we propose a new strategy for choosing optimal parameters in the Chebyshev-Gegenbauer reconstruction method, specifically to achieve numerical stability. This strategy is based on asymptotic analysis as well as minimization problems in one and two dimensions. The effectiveness of our approach, which could also be applied to a wider selection of polynomials is then illustrated with results from numerical experiments.