Parameter optimization and reduction of round off error for the Gegenbauer reconstruction method

The Gegenbauer reconstruction method has been successfully implemented to reconstruct piecewise smooth functions by both reducing the effects of the Gibbs phenomenon and maintaining high resolution in its approximation. However, it has been noticed in some applications that the method fails to converge. This paper shows that the lack of convergence results from both poor choices of the parameters associated with the method, as well as numerical round off error. The Gegenbauer polynomials can have very large amplitudes, particularly near the endpoints x= +/- 1, and hence the approximation requires that the corresponding computed Gegenbauer coefficients be extremely small to obtain spectral convergence. As is demonstrated here, numerical round off error interferes with the ability of the computed coefficients to decay properly, and hence affects the method's overall convergence. This paper addresses both parameter optimization and reduction of the round off error for the Gegenbauer reconstruction method, and constructs a viable "black box'' method for choosing parameters that guarantee both theoretical and numerical convergence, even at the jump discontinuities. Validation of the Gegenbauer reconstruction method through a-posteriori estimates is also provided.