基于Matlab的Laplace方程的高阶差分法数值模拟研究

Numerical Simulation of Laplace Equation Based on Matlab by Higher Order Difference Method

利用多项式外插来处理边界条件,提高差分法数值离散的收敛速度.基于有限差分法求解Laplace微分方程的基本原理,通过数值模拟差分法对不同边界条件处理和外插值的收敛速度进行分析.结果表明,当边界点与离散点靠的很近时,用边界点直接代替离散点处的函数值;当边界点与离散点比较远时,用Lagrange插值得到此离散点.外插法数的数值解和解析解十分接近.

Polynomial extrapolation is used to deal with boundary conditions,and the convergence speed of numerical discretization of difference method is improved.Based on the basic principle of solving Laplace differential equations by finite difference method,the convergence rate of different boundary conditions and extrapolation is analyzed by numerical simulation difference method.The results show that when the boundary point is very close to the discrete point,the function value at the discrete point is directly replaced by the boundary point,when the boundary point is far from the discrete point,the discrete point is obtained by Lagrange interpolation.The numerical solution of the extrapolation method is very close to the analytical solution.