## 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(vt − a(t, x)v
_{xx} + b(t, x)v
_{x} + r(t, x)v, v − ϕ(t, x)) = f(t, x). For the scheme building on the second-order BDF formula, 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.

Original language | English |
---|---|

Journal | I M A Journal of Numerical Analysis |

Volume | 41 |

Issue number | 2 |

Pages (from-to) | 900-934 |

ISSN | 0272-4979 |

DOIs | |

Publication status | Published - Apr 2021 |

## Keywords

- Backward differentiation formula
- Crank–Nicolson scheme
- Diffusion equation
- Finite difference scheme
- High-order schemes
- Numerical methods
- Obstacle equation
- Viscosity solution