Advanced Numerical Methods for Wave Eqn?

[Bork]

Hi guys,

I'm finishing up a term project in a numerical analysis course, and one of the last things left that I need to do is provide a brief survey of the methods used by modern researchers to solve the 2-D wave equation initial boundary value problem. My group solved this equation for an octagonal boundary using a basic numerical discretization method, converting it into a system of ordinary differential equations in time and then solving this system by several numerical methods for ODE's.

I'm looking for some info on methods used in the real world by modern researchers, and I haven't had much luck so far. Obviously there's the separation of variables method, but that's rarely used except when the problem has a convenient symmetry. I'm also aware of the Green's function method, which I presume is easy to implement by converting the problem into one of numerical integration. So my question is whether anyone can point me towards some resources that would explain what's being done these days for the 2-D wave equation IBVP. I don't need to go into too much depth or detail as I'm only asked to do a brief survey of the literature and to present it at an undergraduate level.

Any help would be greatly appreciated!

 

[jvicens]

Hi there,

There are several methods that modern researchers use to solve the 2-D wave equation initial boundary value problem. One of the most commonly used methods is the finite difference method, which is similar to the basic numerical discretization method that your group used. This method involves discretizing the equation into a grid and using numerical algorithms to solve the resulting system of equations.

Another commonly used method is the finite element method, which involves dividing the domain into smaller elements and using piecewise polynomial functions to approximate the solution. This method is often used for complex geometries and can provide more accurate results compared to the finite difference method.

In addition to these methods, researchers also use spectral methods, which involve representing the solution as a sum of basis functions and solving the resulting system of equations using techniques from Fourier analysis. This method is often used for problems with smooth solutions and can provide high accuracy.

As you mentioned, the separation of variables method is also used in some cases, particularly when the problem has a convenient symmetry. This method involves separating the solution into simpler components and solving each component separately.

Other methods that are used by modern researchers include the boundary element method, the method of characteristics, and the method of fundamental solutions. These methods all have their own strengths and weaknesses, and the choice of method often depends on the specific problem being solved.

I recommend checking out some textbooks or research papers on numerical methods for partial differential equations, as they often provide a comprehensive overview of the different methods used for the 2-D wave equation IBVP. Some resources that may be helpful include "Numerical Methods for Partial Differential Equations" by George F. Pinder and "Finite Element Methods for Wave Equations" by J. Tinsley Oden and Leszek F. Demkowicz.

I hope this helps with your survey and good luck with your term project!