求解二阶线性常微分方程的一个显式差分格式

An Explicit Difference Scheme to Solve A Second Order Linear Ordinary Differential Equation

在求解常微分方程的方法中,有限差分法是使用最广泛的方法之一。考虑一个二阶线性常微分方程的边值问题,利用有限差分法,建立了一个具有二阶精度的显式差分格式。首先,通过讨论该显式差分格式的系数矩阵,证明了该显式差分格式解的存在性。然后,通过定义的3种不同范数之间的关系,证明了显式差分格式解的收敛性和稳定性。最后,通过计算机编程对实例的计算,验证了该显式差分格式的数值结果具有二阶精度,并且该显式格式数值结果绘制的图形稳定、光滑,与解析结果吻合较好。

Finite difference method is the one of the most widely used methods to solve ordinary differential equations. This paper conceives a boundary value problem of a second-order linear ordinary differential equations. By using the finite difference method, an explicit difference scheme with second-order accuracy has been set up. First, by discussing the coefficient matrix of the explicit difference scheme, we obtain the existence of solutions. Then, we define three norms in this paper to prove the convergence and stability of the solution. Finally, we apply the method to a concrete example with the help of computer, which verifies that the numerical results occupy the second-order accuracy. The graphic given by the numerical results is stable, smooth and in good agreement with the analytical results.