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I have a system of z-propagated nonlinear PDEs that I solve numerically via a pseudo-spectral method which incorporates adaptive step size control using a Runge-Kutta-Fehlberg technique. This approach is fine over short propagation lengths but computation times dont scale well with propagation distance.

My question is are there any special approaches for solving PDEs over very large domains? If so can these be applied to NL equations? I dont require exact solutions just some idea of the long range dynamics.

Thanks for reading, more information can be supplied if what I have asked isn't clear. I also apologise if I just asked the mathematical equivalent of turning lead into gold, i'm an experimentalist who has accidentally become a theorist, so be kind!

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# Numerical methods for nonlinear PDEs in large domains

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