摘要
对于四阶抛物模型方程周期初值问题,可用有限差分方法进行求解。通常的有限差分方法在使用过程中受到精度和稳定性的限制。本文首先将四阶抛物型方程转化为一个二阶的偏微分方程组,然后对时间项采用子域精细积分的方法、空间项采用三次样条基本公式进行离散,得到了一个含参数α>0(αh)的无条件稳定的差分格式,所得到的差分方程为五点、两层隐格式,它的局部截断误差为O(2τ+α2τ+h4)。,τh分别为时间及空间步长,最后的数值实验表明,本文的方法具有很好的数值精度和良好的实用性。
At present, some researchers have got a lot of good numerical solutions for the periodic initial value problem of four order parabolic equation： such as finite difference method, finite elements method, spectral Galerkin method and so on. Of the methods, the finite difference method is used mostly. However, the method is restricted with numerical stability and precision in the course of using it. However, sometimes the method is not conducive to solve the practical problem well. So given unconditional a stable and high precision method is of great significance. And many experts and scholars are researching it all the time. In this paper based on sub-domain precise integration method, an unconditional stable sub-domain precise integration implicit scheme containing parameter α〉0（α〈〈h） is presented. Rubin, a foreign expert, put forward the cubic spline collocation for numerical solutions of partial differential equation first. And some people have been researching collocation for numerical solutions partial differential equation from that time on. They have given a 3 × 3 matrix system which can be got solution directly. They generalize the alternating direction implicit of difference method to spline function. At same time they research high precise computation scheme. Being flowers, some foreign and civil experts have further developed the spline collocation method. They have got some good results. But cubic spline Collocation being used in periodic initial value problem in four order parabolic equation has not example so far. Therefore, this paper uses sub-domain precise integration idea and cubic spline collocation method to solve initial value problem of four order parabolic equation. It brings forth a new idea. The paper is arranged as follows in detail： First, the parabolic equation is transferred to second order partial differential equation set; next, the equation is here discrete by using sub-domain precise integration method for time and using cubic spline basic formula for space. And the difference equation can be solved by the method of forward elimination and backward substitution. The stability condition of the spline sub-domain precise integration method system is discussed, the local truncation error is O（τ^2＋ατ^2＋h^4）, where τ and h represent the time step and space step respectively. The method does not need iterative method matrix computing. The total in number or amount of calculation is small. To solve the problem in this way is thought conveniently. The resuh shows the collocation method in this paper is better than the method of Saul＇ev, and it＇s precision is higher than Crank-Nicholson classical scheme. This scheme in this paper is efficiency and practicable. The numerical example at the end of this paper is given. It has shown that the numerical result of practical computing accords with the theoretic analysis. The method is of much practical value.
出处
《重庆师范大学学报（自然科学版）》
CAS
2007年第4期33-36,共4页
Journal of Chongqing Normal University:Natural Science
基金
广西民族大学研究生教育创新计划(No.gxun-chx0754)
关键词
四阶抛物型方程
子域精细积分
配置法
four order parabolic equation
sub-domain precise integration
collocation method
