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

Abstract

We propose a new monotone finite-difference scheme for the second-order local approximation on a nonuniform 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 c-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