In this article, compact finite difference approximations for first and second derivatives on the non-uniform grid are discussed. The construction of diffusion wavelets using compact finite difference approximation is presented. Adaptive grids are obtained for non-smooth functions in one and two dimensions using diffusion wavelets. High-order accurate wavelet-optimized compact finite difference (WOCFD) method is developed to solve convection–diffusion equations in one and two dimensions on an adaptive grid. As an application in option pricing, the solution of Black–Scholes partial differential equation (PDE) for pricing barrier options is obtained using the proposed WOCFD method. Numerical illustrations are presented to explain the nature of adaptive grids for each case.

中文翻译：

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对流-扩散方程的小波优化紧凑有限差分法

在本文中，讨论了非均匀网格上一阶和二阶导数的紧凑有限差分近似。提出了使用紧凑有限差分近似的扩散小波的构造。使用扩散小波为一维和二维的非光滑函数获得自适应网格。开发了高阶精确小波优化紧凑有限差分 (WOCFD) 方法来求解自适应网格上的一维和二维对流扩散方程。作为期权定价中的一个应用，Black–Scholes 偏微分方程 (PDE) 的定价障碍期权的解是使用所提出的 WOCFD 方法获得的。给出了数值插图来解释每种情况下自适应网格的性质。