Solving the 2D Helmholtz Partial Differential Equation Using Finite Differences

This Demonstration implements a recently published algorithm for an improved finite difference scheme for solving the Helmholtz partial differential equation on a rectangle with uniform grid spacing. Dirichlet and Sommerfeld boundary conditions are supported. You can specify different source functions . You can prescribe Sommerfeld boundary conditions on up to three edges of the rectangle at the same time. You can vary the value and the angle of incidence . The numerical scheme is converted to a standard system of linear equations system, which can then be solved. You can view the generated matrix and its eigenvalues as well as the solution data using the dropdown menu in the top row.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


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 can be viewed using the dropdown menu. The discretized scheme is given by where and .
Click the "solve" button after making changes to the UI variables to get a new solution. The "reset" button is used to initialize the system back to the state it was before the "solve" button was clicked. Different types of plots and options are available to choose from.
[1] Y. S. Wong and G. Li, "Exact Finite Difference Schemes for Solving Helmholtz Equation at Any Wavenumber," International Journal of Numerical Analysis and Modeling, Series B, Computing and Information, 2(1), 2010 pp. 91–108.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+