Abstract
In this paper, a new optimal fourth-order 21-point finite difference scheme is proposed to solve the 2D Helmholtz equation numerically, with the technique of matched interface boundary (MIB) utilized to treat boundary problems. For the approximation of Laplacian, two sets of fourth-order difference schemes are derived firstly based on the Taylor formula, with a total of 21 grid points involved. Then, a weighted combination of the two schemes is employed in order to reduce the numerical dispersion, and the weights are determined by minimizing the dispersion. Similarly, for the discretization of the zeroth-order derivative term, a weighted average of all the 21 points is implemented to obtain the fourth-order accuracy. The new scheme is noncompact; hence, it encounters great difficulties in dealing with the boundary conditions, which is crucial to the order of convergence. To tackle this issue, the matched interface boundary (MIB) method is employed and developed, which is originally used to accommodate free edges in the discrete singular convolution analysis. Convergence analysis and dispersion analysis are performed. Numerical examples are given for various boundary conditions, which show that new scheme delivers a fourth order of accuracy and is efficient in reducing the numerical dispersion as well.
Highlights
A Fourth-Order 21-Point Finite Difference SchemeWe propose a new fourth-order 21-point finite difference scheme for solving the 2D Helmholtz equation and present the convergence analysis
Introduction eHelmholtz equation, known as frequency-domain wave equation, is widely applied in many areas of science and engineering, such as in aviation, marine technology, geophysics, and optics. e Helmholtz equation is so important that its numerical simulation has attracted significant research interest
We focus on two key points. e first one is to construct a noncompact difference scheme based on minimizing the numerical dispersion, so as to improve the numerical accuracy efficiently
Summary
We propose a new fourth-order 21-point finite difference scheme for solving the 2D Helmholtz equation and present the convergence analysis. We use the classic fourth-order central difference formulas to approximate z2u/zx. For this end, let b1, b2, and b3 be another group of weight parameters which satisfy b1 + b2 + b3 1. We obtain a new 21-point finite difference scheme for 2D Helmholtz (1) by combining (8), (9), and (19) as follows:. There are three sets of discretization schemes for approximating the zeroth-order term k2u, which use 21 grid points. E 21-point finite difference scheme (20) is of fourth order in accuracy. Substituting (35)–(37) into (16) and applying the Taylor formula again, we obtain. Which shows that the new scheme is of fourth order in accuracy
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