基于多项式插值的有限差分法求解Helmholtz方程声硬散射体散射问题

Finite Difference Method Based on Polynomial Interpolation for Solving Hard Acoustic Scattering Problems of the Helmholtz Equation

摘要: 有限差分公式在无网格方法求解微分方程数值解中起着重要作用。本文针对Helmholtz方程的声硬散射体散射问题，通过多项式插值来创建有限差分公式。本文运用一种简单实用的节点分布，既保证多元多项式插值的唯一可解性，又使矩阵为三角矩阵，以便构造的基本多项式化为Lagrange基多项式。最后给出了Helmholtz方程Neumann问题的数值算例。

Abstract: The finite difference formula plays an important role in solving the numerical solution of differential equations by the meshless method. In this paper, the finite difference formula is created by polynomial interpolation for the scattering problem of acoustic hard scatterers of the Helmholtz equation. We use a simple and practical node distribution, which not only ensures the unique solvability of multivariate polynomial interpolation, but also makes the matrix a triangular matrix, so that the basic polynomial constructed can be transformed into Lagrange basis polynomial. Finally, we give the numerical example of the Neumann problem of the Helmholtz equation.