Cranck-Nicolson method for solving hyperbolic PDE?

In summary, the question asks if the Crank-Nicolson method can be used to solve a wave equation with a source term. The person has already solved it with forward euler but the time step has to be small for stability and is looking for a more efficient method. The Crank-Nicolson method is usually used for parabolic PDEs, but the hyperbolic case is more difficult and may require a different method. A link is provided for further reading on solving PDEs with finite difference methods.
  • #1
Crispus
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Crank-Nicolson method for solving hyperbolic PDE?

Hi. I'm not really sure if this is the right part of the forum to ask since it's not really a home-work "problem". Anyway, the question seemed too trivial to ask in the general math forum.


What I'm wondering is wether the Crank-Nicolson method can be used with this PDE (wave equation with a source term i believe):

d2u/dt2 = c^2 * d2u/dx2 + f(t)

f(t) is known for all t.
c is constant.

I have solved the problem with forward euler but the time step has to be really small to have stability. There are actually eight equations to solve so it takes really long time. Solving one of them (using matlab) took 6 min on my computer (2.6GHz).
So I'd like to use an implicit method. And the only one I know of is the Crank-Nicolson method.

The only info I have found about the Crank-Nicolson method in textbooks or on the internet only covers the heat-flow equation.

Can I use normal central difference method for approximation of the d2u/dt2 and then proceed as normal or can this cause stability problems?
(As normal = http://sepwww.stanford.edu/sep/prof/bei/fdm/paper_html/node15.html)
 
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  • #2
No, for efficient (i.e. accurate and fast) solution you need different methods for elliptic, parabolic, and hyperbolic PDEs. Crank-Nicholson is a good method for the parabolic case (e.g. heat conduction, diffusion). The hyperbolic case is hard one to solve efficiently (otherwise, computational fluid dynamics would be easy!)

This link may help - explore a bit to find the best place to start reading, depending on what you know already: http://math.fullerton.edu/mathews/n2003/FiniteDifferencePDEMod.html
 
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  • #3


Thanks for your help!

Hi there,

As a fellow scientist, I can assure you that the Crank-Nicolson method can indeed be used to solve hyperbolic PDEs, including the wave equation with a source term. The Crank-Nicolson method is a popular and widely used numerical method for solving time-dependent PDEs, and it is known for its stability and accuracy.

In order to use the Crank-Nicolson method for your specific PDE, you will need to discretize both the time and space variables using central difference approximations, as you have mentioned. This will result in a system of equations that can be solved using standard matrix methods. The source term, f(t), can be easily incorporated into the system of equations.

One important consideration when using the Crank-Nicolson method is the choice of time step. While this method is known for its stability, it still requires a small enough time step to accurately capture the dynamics of the system. So, you may still need to experiment with different time step sizes to find the optimal one for your problem.

I hope this helps and good luck with your research!
 

What is the Cranck-Nicolson method?

The Cranck-Nicolson method is a numerical method used for solving hyperbolic partial differential equations (PDEs). It is a combination of the forward-time and backward-time finite difference methods, and is known for its ability to accurately capture the behavior of the solution over long time intervals.

How does the Cranck-Nicolson method work?

The Cranck-Nicolson method works by discretizing the PDE into a set of algebraic equations, which can then be solved using matrix algebra techniques. It uses a time-stepping approach, where the solution at each time step is calculated based on the solution at the previous time step. This method is implicit, meaning that it takes into account information from both the past and future time steps.

What are the advantages of using the Cranck-Nicolson method?

One of the main advantages of the Cranck-Nicolson method is its stability and accuracy. It is known to produce more accurate solutions compared to other numerical methods, such as the forward-time or backward-time methods. It is also unconditionally stable, meaning that it can handle a wide range of initial conditions without the solution exploding or becoming unstable.

Are there any limitations to the Cranck-Nicolson method?

While the Cranck-Nicolson method is a powerful numerical method, it does have some limitations. One limitation is that it is computationally more expensive compared to other methods, which can make it less practical for certain applications. It is also more difficult to implement compared to other methods, as it involves solving a system of equations at each time step.

In what applications is the Cranck-Nicolson method commonly used?

The Cranck-Nicolson method is commonly used in various fields of science and engineering, such as fluid dynamics, heat transfer, and quantum mechanics. It is also used in weather forecasting and in the simulation of wave propagation in various media. It is a versatile method that can be applied to a wide range of problems involving hyperbolic PDEs.

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