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This Demonstration implements a recently published algorithm of an improved finite difference scheme for solving the Helmholtz differential equation

in one dimension. Dirichlet and Sommerfeld boundary conditions are supported. Different source functions

can be specified. Sommerfeld boundary conditions can be specified at either end of the domain but not at both ends at the same time. The current value of points per wavelength (PPW) is shown at the top of the display. You can also vary the

value. You can view the matrix

and its eigenvalues using the dropdown menu in the top row.

Contributed by: Nasser M. Abbasi

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Details of the algorithm are described in [1]. This implementation converts the finite difference scheme to the standard

form and uses the built-in Mathematica function LinearSolve to obtain the solution. Sparse matrices are used. The matrix

and its eigenvalues and the numerical solution vector

can be viewed using the dropdown menu. The table below summarizes the boundary conditions, with

and

the values at the left and right boundaries,

the length of the domain, and

and

the solution at the left and right edge of the domain, respectively.

The PDE

is discretized using

, where

. The points per wavelength (PPW) number is calculated using

Reference

[1] Y. S. Wong and G. Li, "Exact Finite Difference Schemes for Solving Helmholtz Equations at Any Wavenumber," International Journal of Numerical Analysis and Modeling B, 2(1), 2010 pp. 91–108.

Helmholtz Differential Equation (Wolfram MathWorld)

Nasser M. Abbasi

"Solving the 1D Helmholtz Differential Equation Using Finite Differences"
http://demonstrations.wolfram.com/SolvingThe1DHelmholtzDifferentialEquationUsingFiniteDifferen/
Wolfram Demonstrations Project
Published: June 3, 2012

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Solving the 1D Helmholtz Differential Equation Using Finite Differences

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