二维泊松方程的两步预估校正格式

A TWO-STEP PREDICTIVE-CORRECTIVE SCHEME FOR 2D POISSON EQUATION

为避免用四阶紧致格式求解泊松方程所具有编程复杂和难以实现的困难,对传统的五点二阶中心差分格式进行改进;通过增加对残差的校正计算,提出了一种新型具有四阶精度的两步预估校正格式.新格式虽需要增加一定的计算量,但它的格式精度高,边界条件处理极简单,易于编程实现.通过数值实验,结果证明上述格式的确具有易于编程和计算精度高的优点.预估校正格式很容易推广到其他复杂情形.

Since the classic 4th order compact scheme requires nine points for 2D Poisson equation,the boundary conditions make the scheme difficult to be implemented,especially when the equations need to be solved by a multi-grid method.A new 4th order accurate scheme is derived.First,a predictive solution is obtained by using the classic 2nd order five-point scheme.Then,the solution is corrected by considering the residual term. Finally,the scheme combines the predictive and the corrective solutions to achieve 4th order accuracy.In each step,the classical 2nd order scheme is used and the boundary conditions are easy to be treated.Although the scheme costs twice of the computational time as compared with the 2nd order scheme,it is much more accurate than the 2nd order scheme.Moreover,the boundary conditions are quite simple and it is easier to code for this new scheme.The numerical simulations confirm that the new scheme has the same accuracy as the classic 4th order compact scheme.This provides a new approach to turn a lower order scheme in to a high order scheme.