基于不光滑边界的变系数抛物型方程的高精度紧格式

本文讨论基于不光滑边界的变系数抛物型方程求解的高精度紧格式.首先构造一般变系数抛物型方程的高精度紧格式,并在理论上证明格式具有空间方向四阶精度.然后针对非光滑边界条件,引入局部网格加密技巧在奇异点附近进行不均匀的网格加密.数值实验以期权定价中Black-Scholes偏微分方程的求解为例,验证高精度紧格式用于光滑边界条件的微分方程离散可以达到四阶精度.对于处理非光滑边界条件,网格局部加密技巧能有效的提高数值解精度,使得高精度紧格式用于定价欧式期权可以接近四阶精度.

HIGH ORDER COMPACT SCHEMES FOR VARIABLE COEFFICIENT PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS WITH NON-SMOOTH BOUNDARY CONDITIONS

For the solution of variable coefficient parabolic partial differential equations, the general high order compact schemes which are analyzed to be fourth order accuracy theoretically are proposed, and the local mesh refinement strategy is implemented in space in order to overcome the non-smoothness of the boundary conditions. Numerical examples on Black-Scholes European options pricing with smooth boundary conditons further demonstrate the fourth order accuracy of the proposed schemes, and for the non-smooth case, the proposed scheme with local mesh refinement strategy is shown to attain high order accuracy.