Numerical method for wave equation

In summary, the speaker is trying to plot a function that is subject to a nonlinear wave equation. They have found the split step Fourier method as a potential solution for the nonlinear Schrodinger equation, but it only works for a specific form of PDE with an analytic solution. The speaker is wondering how to solve numerically for any arbitrary PDE, specifically the one given in the conversation. They are looking for a general method and do not need boundary conditions for this purpose. They ask for an explanation of the split step Fourier method without knowledge of the boundary conditions.
  • #1
semc
368
5
Hi, I am trying to plot a function subjected to a nonlinear wave equation. One of the method I found for solving the nonlinear schrodinger equation is the split step Fourier method. However I noticed that this method only works for a specific form of PDE where the equation has an analytic solution for both the linear and nonlinear part. So how do I solve numerically any arbitrary PDE? Specifically, my PDE is

[itex]\partial_zE(r_⊥,z,\tau)-∇^2_⊥\int^{\tau}_{-\infty}d\tau 'E(r_⊥,z,\tau ')=\int^{\tau}_{-\infty}d\tau '\omega^2(r_⊥,z,\tau ')-\frac{\partial_{\tau}n(r_⊥,z,\tau)}{E(r_⊥,z,\tau)}[/itex]

and I want to solve for the evolution of E. Any help would be greatly appreciated. Thanks!
 
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  • #2
Boundary conditions, please.
 
  • #3
From my experience with PDE, you only need boundary conditions when solving for specific solutions. I am looking for a way to solve the PDE numerically. Would it be possible to give me the general method? For example you can [itex]\bf{explain}[/itex] the split step Fourier method without knowing the boundary conditions?
 

FAQ: Numerical method for wave equation

What is a numerical method for solving the wave equation?

Numerical methods for solving the wave equation are mathematical algorithms that use numerical calculations to approximate the solution of the wave equation. These methods are used when an analytical solution cannot be found or is too complex to solve.

How does a numerical method for solving the wave equation work?

A numerical method for solving the wave equation works by dividing the wave equation into smaller, discrete parts and using numerical techniques, such as finite difference or finite element methods, to approximate the solution at each point. These solutions are then combined to approximate the overall solution of the wave equation.

What are the advantages of using a numerical method for solving the wave equation?

One advantage of using a numerical method for solving the wave equation is that it can be used to solve complex problems that do not have an analytical solution. It also allows for the use of computers to quickly and accurately calculate the solution. Additionally, numerical methods can handle a wide range of boundary conditions and can be adapted for different types of wave equations.

What are the limitations of numerical methods for solving the wave equation?

One limitation of numerical methods for solving the wave equation is that they are only approximations and may not provide an exact solution. The accuracy of the solution also depends on the chosen numerical method and the size of the discretized parts. Additionally, these methods can be computationally demanding and may require advanced programming skills to implement.

What are some practical applications of numerical methods for solving the wave equation?

Numerical methods for solving the wave equation have many practical applications in various fields, including acoustics, electromagnetics, and structural dynamics. They are used to model and predict the behavior of waves in different mediums, such as sound waves in air, electromagnetic waves in communication systems, and seismic waves in the Earth's crust. These methods are also used in engineering and design to analyze the response of structures to different types of waves.

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