Backward Differentiation Formula finite difference schemes for diffusion equations with an obstacle term

TL;DR

This paper investigates BDF finite difference schemes for one-dimensional diffusion equations with obstacle terms, establishing stability, error estimates, and second-order convergence, with applications in finance and numerical analysis.

Contribution

It introduces and analyzes BDF2 schemes for obstacle problems, proving stability, error bounds, and demonstrating second-order convergence both theoretically and numerically.

Findings

BDF2 scheme is unconditionally stable for obstacle problems.

The scheme achieves second-order convergence in space and time.

Numerical tests confirm theoretical error estimates and applicability to finance problems.

Abstract

Finite difference schemes, using Backward Differentiation Formula (BDF), are studied for the approximation of one-dimensional diffusion equations with an obstacle term, of the form $$\min(v_t - a(t,x) v_{xx} + b(t,x) v_x + r(t,x) v, v- \varphi(t,x))= f(t,x).$$ For the scheme building on the second order BDF formula (BDF2), we discuss unconditional stability, prove an $L^2$-error estimate and show numerically second order convergence, in both space and time, unconditionally on the ratio of the mesh steps. In the analysis, an equivalence of the obstacle equation with a Hamilton-Jacobi-Bellman equation is mentioned, and a Crank-Nicolson scheme is tested in this context. Two academic problems for parabolic equations with an obstacle term with explicit solutions and the American option problem in mathematical finance are used for numerical tests.