The best method to solve Helmholtz equation for a irregular boundary

In summary, the conversation discusses solving the Helmholtz equation with a modified boundary condition on an almost square region. The individual is unsure of the best numerical method and asks about the use of Finite element, a method they are not familiar with. A suggestion is made to use Taylor series expansions at the boundary to account for the difference between the actual boundary and the square boundary, allowing for the use of finite differences. A reference to a book on applied numerical methods is also mentioned.
  • #1
wdlang
307
0
i have an almost square region.

By 'almost' i mean the edges are curvy, not completely straight.

i now need to solve the Helmholtz equation with Dirichlet boundary condition

what is the best numerical method?

how is Finite element, though i do not know what Finite element is
 
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  • #2
By proper use of Taylor series expansions at the boundary, you can express a modified boundary condition on a square boundary (taking into account the difference between the actual boundary location and the square boundary location). This will allow you to use finite differences with the square boundary. See Carnahan, Luther, and Wilkes, Applied Numerical Methods.
 

Related to The best method to solve Helmholtz equation for a irregular boundary

1. What is the Helmholtz equation and why is it important?

The Helmholtz equation is a partial differential equation that describes the behavior of waves in a variety of physical systems, including acoustics, electromagnetism, and quantum mechanics. It is important because it allows us to understand and predict the behavior of waves in complex systems, which has numerous applications in engineering and science.

2. What do you mean by an irregular boundary in the context of the Helmholtz equation?

An irregular boundary refers to a boundary that is non-uniform or non-smooth. This could include boundaries with sharp corners, varying curvature, or other irregularities. In the context of the Helmholtz equation, an irregular boundary poses a challenge for finding an analytical solution, and numerical methods are often needed to solve the equation.

3. What are some common numerical methods used to solve the Helmholtz equation for irregular boundaries?

Some common numerical methods used to solve the Helmholtz equation for irregular boundaries include finite difference methods, finite element methods, and boundary element methods. These methods discretize the domain and approximate the solution at discrete points, allowing for the solution to be calculated numerically.

4. Are there any limitations to using numerical methods for solving the Helmholtz equation for irregular boundaries?

Yes, there can be limitations to using numerical methods for solving the Helmholtz equation for irregular boundaries. These methods can be computationally expensive, and the accuracy of the solution can depend on the chosen discretization scheme and the complexity of the boundary. Additionally, some numerical methods may not be well-suited for certain types of irregular boundaries.

5. How can the accuracy of a numerical solution for the Helmholtz equation be improved for an irregular boundary?

There are several ways to improve the accuracy of a numerical solution for the Helmholtz equation with an irregular boundary. One approach is to use a more refined discretization, such as a higher order finite element or finite difference method. Another option is to use adaptive meshing techniques, which can adjust the discretization based on the solution behavior near the boundary. Additionally, incorporating analytical or semi-analytical solutions for simpler boundary geometries can also improve the accuracy of the numerical solution.

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