Backward differentiation formula finite difference schemes for diffusion equations with an obstacle term
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.
Origin : Files produced by the author(s)
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