ABSTRACT

A brief introduction is given of the patterns and structures of the residue appears in the defect correction framework for partial differential equations. The concept is illustrated through a model problem with a small parameter such that its solution exhibits asymptotic behaviour. The idea of automated asymptotic expansion is discussed at this point. A high order option valuation PDE derived from the Black-Scholes equation with a nonlinear volatility function is used to demonstrate the automated asymptotic expansion. Numerical solutions are compared to the standard Newton’s linearisation method. The advantage of this method is discussed. Finally, some current theoretical work of the present method is discussed.