四阶抛物型方程子域精细积分紧致差分格式

Compact Crank-Nicolson Scheme for Solving Four Order Parabolic Equation Based on Sub-domain Precise Integration

首先给出了四阶导数的紧致差分公式，然后应用子域精细积分的方法，本文构造出了一个求解四阶抛物型方程周期初值问题的含参数α（0〈α〈〈△t）的紧致格式，所得到的差分格式为五点、两层的隐格式。Fourier分析方法表明该格式为无条件稳定，其局部截断误差为O（α（△t）^2＋α^2（At）^3＋（△x）^4），其中△t，△x分别为时间步长和空间步长，误差分析和数值实验均表明，本文构造的格式比经典的Crank—Nicholson格式和Saul’ev构造的格式精度要高阶10^-3-10^-4。从精度及稳定性方面考虑，本文构造的格式也较好，因此，本文的差分格式是有效的，具有很好的实用性。

Recently Zeng Wen-ping has proposed a factor to be determined, in other words, using a difference equation contained a number of parameters in approximation to the differential equation. These parameters of the equation can be obtained by error analysis. Zeng Wen-ping got many high accuracy difference schemes, however, some of them are three level and conditionally stable difference scheme. It is hard to construct a unconditionally stable and high precision scheme. The classic Crank-Nicholson scheme is unconditionally stable,but its accuracy is too low. At the same time, the Sub-domain Precise Integration method was first introduced by Zhong Wan-xie for solving the partial differential equation in 1995. Later, many scholars utilized the Sub-domain Precise Integration to solve the convection equation and convection-diffusion equation. They obtained a lot of unconditionally stable difference schemes. Very recently parabolic equation is constructed based on the Sub-domain Precise Integration method in time direction. The difference scheme is five-point and two level implicit scheme. The coefficient matrix of this difference equation is the strict diagonally dominant, it can be solved by the square root. Stability analysis of this scheme has been carried out by Fourier. It is shown that this scheme is unconditionally stable,and the local truncation error is O（α（△t）^2 ＋ α^2（△t）^3 ＋ （△x）^4） o It is shown by both error analysis and numerical examples that the accuracy of the present method is much better than the classical Crank-＇Nicholson method and Saul＇ev method in[ 1 ], and the accuracy and stability of the present method is much better than the explicit scheme in [ 5 ]. Therefore, the difference format is effective and has a good Practicality. The numerical experiments at the end of this paper have shown that the numerical results are in agreement with the theoretical analysis.