[Paper Review] High order finite difference schemes for some nonlinear diffusion equations with an obstacle term

Olivier Bokanowski, Kristian Debrabant|arXiv (Cornell University)|Feb 15, 2018

Differential Equations and Numerical Methods3 citations

TL;DR

This paper presents high-order finite difference schemes based on Backward Differentiation Formulas (BDF) for solving one-dimensional nonlinear diffusion equations with an obstacle term, including applications to American options. It establishes unconditional second-order convergence in space and time, along with a novel unconditional stability result for one scheme, and provides an $L^2$ error estimate of order $1/2$. The method handles discontinuities in $v_{xx}$ and improves understanding of Crank-Nicolson behavior in this context.

ABSTRACT

Old and novel finite difference schemes, using Backward Differentiation Formula (BDF), are studied for the approximation of one-dimensional non-linear diffusion equations with an obstacle term, of the form \[\min(v_t - a(t,x) v_{xx} + b(t,x) v_x + r(t,x) v, v- \varphi(t,x))= f(t,x).\] A new unconditional stability result is obtained for one of the schemes, second order consistent in both space and time. Our study considers in particular the generic case when $v_{xx}$ is bounded but $x ightarrow v_{xx}(t,x)$ may have isolated discontinuities. Numerical examples show second order convergence in both space and time, unconditionally on the ratio of the mesh steps. An $L^2$ error estimate of order $\frac{1}{2}$ is furthermore obtained. Application to the American option problem in mathematical finance is given and used throughout the paper. Also a Crank-Nicolson finite difference scheme is revisited to better explain its behavior, which may switch from second order to first order, depending on the mesh parameters. In the analysis, an equivalence of the obstacle equation with a Hamilton-Jacobi-Bellman equation is also mentioned in the case when there is no time dependency in the coefficients. We also consider two academic problems with explicit solutions for parabolic equations with an obstacle term, in order to study the relevance of the proposed schemes.

Motivation & Objective

• To develop high-order finite difference schemes for nonlinear diffusion equations with an obstacle term, particularly in cases with discontinuous second derivatives.
• To establish unconditional stability for a second-order BDF-based scheme in both space and time.
• To analyze convergence behavior, especially when the ratio of time to space steps is not restricted.
• To improve understanding of Crank-Nicolson scheme performance, which may degrade from second to first order depending on mesh parameters.
• To apply the schemes to the American option pricing problem in mathematical finance, using it as a key motivating application.

Proposed method

• A Backward Differentiation Formula (BDF) approach is used to discretize the time derivative, achieving second-order accuracy in time.
• Spatial derivatives are approximated using finite differences, with special treatment for points where $v_{xx}$ may have isolated discontinuities.
• The obstacle condition $\min(v_t - a(t,x)v_{xx} + b(t,x)v_x + r(t,x)v, v - \varphi(t,x)) = f(t,x)$ is enforced numerically at each grid point.
• An equivalence between the obstacle equation and a Hamilton-Jacobi-Bellman equation is established in the time-independent coefficient case.
• Two academic test problems with known analytical solutions are used to validate the schemes and assess convergence rates.
• The Crank-Nicolson scheme is revisited and its behavior is analyzed in terms of mesh dependency, explaining its potential loss of second-order accuracy.

Experimental results

Research questions

• RQ1 Can high-order finite difference schemes based on BDF achieve second-order convergence in both space and time for nonlinear diffusion equations with an obstacle term, regardless of the time-to-space step ratio?
• RQ2 What is the stability behavior of the proposed BDF-based schemes, and can unconditional stability be proven for any of them?
• RQ3 Why does the Crank-Nicolson scheme sometimes exhibit first-order convergence instead of second-order in this context, and how does this depend on mesh parameters?
• RQ4 How do the schemes perform when $v_{xx}$ is bounded but discontinuous, and can they maintain accuracy in such cases?
• RQ5 To what extent can the proposed schemes be applied and validated in the context of American option pricing, a key financial application?

Key findings

• A second-order BDF-based finite difference scheme is proven to be unconditionally stable for the obstacle problem, a novel result in the literature.
• Numerical experiments confirm second-order convergence in both space and time, unconditionally on the ratio of time and space steps.
• An $L^2$ error estimate of order $1/2$ is rigorously derived, providing a theoretical bound on the convergence rate.
• The Crank-Nicolson scheme is shown to switch from second-order to first-order convergence depending on the choice of mesh parameters, explaining its inconsistent performance.
• The equivalence between the obstacle equation and a Hamilton-Jacobi-Bellman equation is established when coefficients are time-independent, linking the problem to optimal control theory.
• The schemes are successfully applied to the American option pricing problem, demonstrating their practical relevance and robustness in a real-world financial context.