Methodology to Compute Mathieu Functions for Arbitrary Large Parameter q and Its Application to Pressure Transient Analysis

Mathieu functions are widely used for the solution of boundary value problems in elliptical systems. In spite of their common use, they are notorious for their inherent instability at high (absolute) values of the parameter q. In this paper we present robust solution methodologies enabling the computation of Mathieu functions for values of q that can go to infinity. First, we present a methodology dealing with inherent instability in the recurrence relationships for Fourier coefficients, thus enabling their accurate computation for arbitrarily high q. Secondly, we overcame the ‘subtraction error’ in the computation of Mathieu functions as infinite sums of Bessel function products by defining asymptotic approximations for ratios between Mathieu function values for different value of the elliptical coordinate. The accuracy of these asymptotic approximations was extensively tested over large ranges of the relevant parameters, and excellent agreement was found. Thirdly, we computed accurate early-time transient pressure profiles for particular sets of boundary conditions by expressing these as linear combinations of Mathieu function ratios (instead of Mathieu functions directly). We illustrate our methodology by applying it to two well-known problems in the area of Pressure Transient Analysis, the limitations of some concepts that are well-accepted in literature are demonstrated.