Finite-difference method for the Gamma equation on non-uniform grids

Le Minh Hieu^1*, Truong Thi Hieu Hanh¹, Dang Ngoc Hoang Thanh²

¹University of Economics, The University of Danang

²Hue College of Industry

Received 1 August 2019; accepted 11 November 2019

 

Abstract:

We propose a new monotone finite-difference scheme for the second-order local approximation on a non-uniform grid that approximates the Dirichlet initial boundary value problem (IBVP) for the quasi-linear convection-diffusion equation with unbounded nonlinearity, namely, for the Gamma equation obtained by transformation of the nonlinear Black-Scholes equation into a quasilinear parabolic equation. Using the difference maximum principle, a two-sided estimate and an a priori estimate in the -norm are obtained for the solution of the difference schemes that approximate this equation.

Keywords: Gamma equation, maximum principle, monotone finite-difference scheme, non-uniform grid, quasi-linear parabolic equation, scientific computing, two-side estimates.

Classification number: 1.1