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 $$\begin{equation*}\min(v_t - a(t,x) v_{xx} + b(t,x) v_x + r(t,x) v, v- \varphi(t,x))= f(t,x).\end{equation*}$$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.

中文翻译：

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具有障碍项的扩散方程组的向后微分方程有限差分格式

摘要

研究了使用后向微分公式（BDF）的有限差分方案，用于近似一维扩散方程，其障碍项的形式为$$ \ begin {equation *} \ min（v_t-a（t，x）v_ {xx} + b（t，x）v_x + r（t，x）v，v- \ varphi（t，x））= f（t，x）。\ end {equation *} $$用于方案构建在二阶BDF公式上，我们讨论了无条件稳定性，证明了$ L ^ 2 $误差估计，并在数值上显示了在空间和时间上无条件地基于网格步长比的二阶收敛性。在分析中，提到了带有汉密尔顿-雅各比-贝尔曼方程的障碍方程的等价关系，并在这种情况下测试了Crank-Nicolson方案。