Introduction to the hyperasymptotic technique in high-frequency computational electromagnetics

Application of integral equation methods in computational electromagnetics has been widely explored in problems with varying degrees of complexity. Central to the formulation of such integral equation methods is the appropriate Green´s function that in most cases is an important contributor to the accuracy of the final solution. It is however well known that at high frequencies special analytical forms of the problem-matched Green´s function reduces the computation resources and hence renders solutions to electrically large problems practicable. These special high-frequency representations are derived analytically by well-known asymptotic methods when the characteristic wavenumber |k| → ∞. In this presentation a novel asymptotic method, known as hyperasymptotics, originally developed by Berry and Howls, is introduced. The main feature of the hyperasymptotic technique is that the numerical error in neglecting the remainder, obtained after optimal truncation of the asymptotic series, is of the order O(e^−C|k|), where C is a positive constant. Thus the error in the hyperasymptotic method decreases exponentially at high frequencies for |k| → ∞, and hence this specific asymptotic technique appears numerically most suitable for development of hybrid methods for challenging problems in computational electromagnetics. The salient features of the hyperasymptotic method is illustrated here with reference to the Stokes phenomenon for the Airy function of complex argument, and, its potential applications to some problems in computational electromagnetics are identified.