Cranck-Nicolson method for solving hyperbolic PDE?

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SUMMARY

The Crank-Nicolson method is primarily designed for parabolic partial differential equations (PDEs) such as heat conduction, and its application to hyperbolic PDEs, like the wave equation with a source term, is not straightforward. The discussion highlights the challenges of using the forward Euler method due to stability issues, particularly when solving multiple equations. While the Crank-Nicolson method can be considered, it is essential to recognize that different numerical methods are required for elliptic, parabolic, and hyperbolic PDEs to ensure efficiency and accuracy in solutions.

PREREQUISITES
  • Understanding of partial differential equations (PDEs)
  • Familiarity with numerical methods for PDEs, specifically Crank-Nicolson and forward Euler methods
  • Knowledge of stability criteria in numerical analysis
  • Proficiency in MATLAB for implementing numerical solutions
NEXT STEPS
  • Research the application of the Crank-Nicolson method to hyperbolic PDEs
  • Explore alternative implicit methods suitable for hyperbolic equations
  • Learn about stability analysis in numerical methods for PDEs
  • Investigate the central difference method for approximating second derivatives in time
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Mathematicians, numerical analysts, and engineers working with PDEs, particularly those interested in solving hyperbolic equations efficiently using numerical methods.

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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)
 
Last edited:
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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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